{"id":37456,"date":"2019-10-30T15:23:26","date_gmt":"2019-10-30T14:23:26","guid":{"rendered":"https:\/\/groupe-reussite.fr\/?page_id=37456"},"modified":"2026-10-27T10:42:24","modified_gmt":"2026-10-27T09:42:24","slug":"cours-en-ligne-suites-numeriques-maths-sup-exercices-corriges","status":"publish","type":"page","link":"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-suites-numeriques-maths-sup-exercices-corriges\/","title":{"rendered":"Suites num\u00e9riques en Maths Sup Exercices corrig\u00e9s"},"content":{"rendered":"<div class=\"Latex_page\"><\/div>\n<div><\/div>\n<div class=\"Latex_page\">\n<section class=\"has_ae_slider elementor-element elementor-section-full_width elementor-reverse-mobile elementor-section-stretched elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default elementor-section elementor-top-section\">\n<div class=\"elementor-container elementor-column-gap-no\">\n<div class=\"elementor-row\">\n<div class=\"elementor-column-wrap elementor-element-populated colone_elementor\">\n<p style=\"text-align: justify;\">\t\t<div data-elementor-type=\"page\" data-elementor-id=\"79298\" class=\"elementor elementor-79298\">\n\t\t\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-top-section elementor-element elementor-element-316e72a0 elementor-section-content-bottom elementor-section-boxed elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"316e72a0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2dff41b5 sticky_badeau_colone strick_col ae-bg-gallery-type-default\" data-id=\"2dff41b5\" data-element_type=\"column\" data-e-type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-top-section elementor-element elementor-element-f93bf96 elementor-hidden-mobile elementor-section-boxed elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"f93bf96\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-912eb3c strick_col ae-bg-gallery-type-default\" data-id=\"912eb3c\" data-element_type=\"column\" data-e-type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2129d29 elementor-widget elementor-widget-heading\" data-id=\"2129d29\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<p class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-maths-sup-maths\/\">Cours en ligne Maths en Maths Sup<\/a><\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7427b38 elementor-widget elementor-widget-heading\" data-id=\"7427b38\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<p class=\"elementor-heading-title elementor-size-default\">Chapitres Maths en MPSI, PCSI, MP2I, PTSI<\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ce583b3 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"ce583b3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-recurrence-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">R\u00e9currence<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b219041 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"b219041\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-sommes-produits-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Sommes et produits<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6efeca2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"6efeca2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-complexes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Nombres complexes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bd49f4c elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"bd49f4c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-trigonometrie-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Trigonom\u00e9trie<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-857a32a elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"857a32a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-nombres-reels-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Nombres r\u00e9els<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6788b8b elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"6788b8b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-ensembles-et-applications-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Ensembles et applications<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-356ef7a elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"356ef7a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-introduction-fonctions-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fonctions (intro)<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c1e84d2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"c1e84d2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fonctions-usuelles-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fonctions usuelles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d684d04 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"d684d04\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-primitives-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Primitives<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a6938f6 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"a6938f6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-equations-differentielles-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">\u00c9quations diff\u00e9rentielles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-276b5ea elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"276b5ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-suites-numeriques-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Suites num\u00e9riques<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f150d4e elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"f150d4e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-limites-et-continuite-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Limites et continuit\u00e9<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d7cf5c0 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"d7cf5c0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-derivees-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9riv\u00e9es<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-600f6e8 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"600f6e8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-systemes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Syst\u00e8mes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-07b2116 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"07b2116\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-polynomes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Polyn\u00f4mes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b8da8fb elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"b8da8fb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fractions-rationnelles-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fractions rationnelles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a439fa3 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"a439fa3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-arithmetique-polynomes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Arithm\u00e9tique et polyn\u00f4mes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3365085 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3365085\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-arithmetique-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Arithm\u00e9tique et fractions rationnelles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ecd9e2f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"ecd9e2f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-analyse-asymptotique-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Analyse asymptotique<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a1dd02f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"a1dd02f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-developpements-limites-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9veloppements limit\u00e9s<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4f5b120 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"4f5b120\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-denombrements-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9nombrements<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-83de7cd elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"83de7cd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-espaces-vectoriels-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Espaces vectoriels<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3b3e27e elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3b3e27e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-matrices-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Matrices<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b32dfbe elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"b32dfbe\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-integration-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Int\u00e9grations<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3df0730 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3df0730\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-prehilbertiens-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Espaces pr\u00e9hilbertiens<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2e83874 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2e83874\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-espaces-euclidiens-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Espaces euclidiens<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bf2a2d5 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"bf2a2d5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-series-numeriques-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">S\u00e9ries num\u00e9riques<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-668487e elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"668487e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-probabilites-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Probabilit\u00e9s<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4a483d2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"4a483d2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-determinants-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9terminants<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e87e186 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"e87e186\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-variables-aleatoires-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Variables al\u00e9atoires<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-23301a1 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"23301a1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/nous-contacter\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">CONTACTEZ-NOUS<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-top-section elementor-element elementor-element-f08add3 elementor-hidden-desktop elementor-hidden-tablet elementor-section-boxed elementor-section-height-default elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"f08add3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bac3e5c 0 ae-bg-gallery-type-default\" data-id=\"bac3e5c\" data-element_type=\"column\" data-e-type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b01bd3f elementor-widget elementor-widget-heading\" data-id=\"b01bd3f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<p class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-maths-sup-maths\/\">Cours en ligne Maths en Maths Sup<\/a><\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a2a7315 elementor-widget elementor-widget-heading\" data-id=\"a2a7315\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<p class=\"elementor-heading-title elementor-size-default\">Chapitres Maths en MPSI, PCSI, MP2I, PTSI<\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ae12357 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"ae12357\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-recurrence-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">R\u00e9currence<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-252c8e5 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"252c8e5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-sommes-produits-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Sommes et produits<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eb5c12c elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"eb5c12c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-complexes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Nombres complexes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3ad63de elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3ad63de\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-trigonometrie-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Trigonom\u00e9trie<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bf9392d elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"bf9392d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-nombres-reels-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Nombres r\u00e9els<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-da2d22b elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"da2d22b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-ensembles-et-applications-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Ensembles et applications<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-72889e2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"72889e2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-introduction-fonctions-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fonctions (intro)<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-29b7df0 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"29b7df0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fonctions-usuelles-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fonctions usuelles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-796e924 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"796e924\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-primitives-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Primitives<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f38e734 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"f38e734\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-equations-differentielles-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">\u00c9quations diff\u00e9rentielles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-09a1510 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"09a1510\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-suites-numeriques-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Suites num\u00e9riques<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-608ac33 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"608ac33\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-limites-et-continuite-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Limites et continuit\u00e9<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-560b787 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"560b787\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-derivees-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9riv\u00e9es<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-252fca2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"252fca2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-systemes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Syst\u00e8mes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d2272da elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"d2272da\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-polynomes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Polyn\u00f4mes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9a4b447 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"9a4b447\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fractions-rationnelles-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Fractions rationnelles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fb59ff6 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"fb59ff6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-arithmetique-polynomes-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Arithm\u00e9tique et polyn\u00f4mes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7b05dc3 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"7b05dc3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-arithmetique-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Arithm\u00e9tique et fractions rationnelles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3838257 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3838257\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-analyse-asymptotique-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Analyse asymptotique<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3bfff8f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3bfff8f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-developpements-limites-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9veloppements limit\u00e9s<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2aaf5e5 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2aaf5e5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-denombrements-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9nombrements<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cc1d27b elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"cc1d27b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-espaces-vectoriels-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Espaces vectoriels<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-57e3056 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"57e3056\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-matrices-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Matrices<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2789857 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2789857\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-integration-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Int\u00e9grations<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3f93649 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3f93649\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-prehilbertiens-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Espaces pr\u00e9hilbertiens<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6b66cee elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"6b66cee\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-espaces-euclidiens-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Espaces euclidiens<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4960203 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"4960203\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-series-numeriques-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">S\u00e9ries num\u00e9riques<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c9f48bd elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"c9f48bd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-probabilites-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Probabilit\u00e9s<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-73f58be elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"73f58be\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-determinants-maths-sup\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9terminants<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-08e6f36 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"08e6f36\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div 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href=\"https:\/\/groupe-reussite.fr\/nous-contacter\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">CONTACTEZ-NOUS<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t\n<\/div>\n<div class=\"elementor-column-wrap elementor-element-populated colone_latex\" style=\"text-align: justify;\">\n<div class=\"style_pag\" style=\"color: #000; font-family: 'Lato', Sans-serif;\">\n<div class=\"elementor-text-editor elementor-clearfix\" style=\"color: #000000; font-family: Lato, sans-serif;\">\n<h1 class=\"Latex_page_title\" style=\"text-align: center;\">Exercices corrig\u00e9s sur les suites num\u00e9riques en maths sup<\/h1>\n<p><em>Par Seda Biber, le 27\/10\/2026<\/em><\/p>\n<\/div>\n<p dir=\"ltr\" style=\"text-align: justify;\">Pour \u00e9tudier une <strong>suite num\u00e9rique en maths sup<\/strong>, commencez toujours par reconna\u00eetre sa forme. Une suite arithm\u00e9tico-g\u00e9om\u00e9trique ou r\u00e9currente lin\u00e9aire d&rsquo;ordre 2 se calcule explicitement. Une suite d\u00e9finie par <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-280c9d22efa1f941d0ba972441a4364d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#102;&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"\/> s&rsquo;\u00e9tudie avec les variations de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/>, le signe de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-22822e890a29961b079db4340fe1e442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"\/> et les intervalles stables. Une suite donn\u00e9e par une somme ou un produit se traite par <strong>encadrement<\/strong>, par <strong>t\u00e9lescopage<\/strong> ou avec une <strong>somme de Riemann<\/strong>. Les 21 exercices corrig\u00e9s ci-dessous couvrent ces m\u00e9thodes, du calcul de limites simples \u00e0 la constante d&rsquo;Euler. Ils suivent le chapitre \u00ab Nombres r\u00e9els et suites num\u00e9riques \u00bb du programme de premi\u00e8re ann\u00e9e en MPSI, MP2I, PCSI et PTSI, et chaque corrig\u00e9 d\u00e9taille la r\u00e9daction attendue en DS comme en colle.<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-suites-numeriques-maths-sup\/\">R\u00e9sum\u00e9 de cours<\/a> | <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-suites-numeriques-maths-sup-exercices-corriges\/\">Exercices et corrig\u00e9s<\/a><\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-maths-sup-maths\/\">Cours en ligne de maths en maths sup<\/a><\/p>\n<\/div>\n<div class=\"style_pag\" style=\"color: #000; font-family: 'Lato', Sans-serif;\">\n<h2>Comment trouver le terme g\u00e9n\u00e9ral d&rsquo;une suite r\u00e9currente ?<\/h2>\n<p>Le programme de maths sup ne demande de savoir calculer explicitement le terme g\u00e9n\u00e9ral que pour quelques familles de suites : arithm\u00e9tiques, g\u00e9om\u00e9triques, arithm\u00e9tico-g\u00e9om\u00e9triques et r\u00e9currentes lin\u00e9aires d&rsquo;ordre 2. Le travail consiste donc souvent \u00e0 se ramener \u00e0 l&rsquo;une d&rsquo;elles par un changement de suite bien choisi (carr\u00e9, inverse, logarithme).<\/p>\n<h3>Exercice 1 : se ramener \u00e0 une suite arithm\u00e9tico-g\u00e9om\u00e9trique<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f51fd3c7b8dce153fab4e0442877b12f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: -3px;\"\/> et, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-588616762f1acc1b153991dd6fa4f83c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0bbff20c54288de1a830c4e0db499454_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#94;&#50;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -13px;\"\/>. Exprimer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> en fonction de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fce4ef575cb9dacb190c93ae8d286cdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"\/> et de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Posons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-87b28faddcc9507391b4212f88ef0e3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#117;&#95;&#110;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -4px;\"\/>. La relation donne <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0349311d451b3fc27efb4809c8d37107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#118;&#95;&#110;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"109\" style=\"vertical-align: -12px;\"\/> : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> est arithm\u00e9tico-g\u00e9om\u00e9trique.<\/p>\n<ul>\n<li>On cherche la constante <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> telle que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9e01b6cf54d6e7abf2b9bf2b1b697f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#67;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"84\" style=\"vertical-align: -12px;\"\/>, soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d481c86b32bd6e5b653830a2f3fcf0d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<li>La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6d0c33c7858186904cf908540a5c59ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#95;&#110;&#61;&#118;&#95;&#110;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"93\" style=\"vertical-align: -3px;\"\/> v\u00e9rifie <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ab638e09e8625f8b86aca271818eea3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#119;&#95;&#110;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"87\" style=\"vertical-align: -12px;\"\/>. Elle est g\u00e9om\u00e9trique de raison <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-49af4d1448907c79187a6244955725a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\"\/> et de premier terme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cee1f1b8c77c1b9f937b251504dede51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#95;&#48;&#61;&#117;&#95;&#48;&#94;&#50;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"92\" style=\"vertical-align: -5px;\"\/>.<\/li>\n<li>Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b9419eb034f9728154b7998f3f21c0ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#48;&#94;&#50;&#45;&#50;&#125;&#123;&#50;&#94;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"96\" style=\"vertical-align: -12px;\"\/> puis <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e1d38935406c3143a04ab5d8f6272729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#50;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#48;&#94;&#50;&#45;&#50;&#125;&#123;&#50;&#94;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"123\" style=\"vertical-align: -12px;\"\/>.<\/li>\n<\/ul>\n<p>Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> est une racine carr\u00e9e, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cb2dfd85874cb39a7152069572f2e210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/> et :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-85093617de5b44cd96f515f060d61cf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#48;&#94;&#50;&#45;&#50;&#125;&#123;&#50;&#94;&#110;&#125;&#125;&#92;&#113;&#117;&#97;&#100;&#40;&#110;&#92;&#103;&#101;&#113;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"217\" style=\"vertical-align: -13px;\"\/><\/p>\n<p>La formule vaut aussi pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6c42dfb3cd9a03af86343d794610e3c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"\/> si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ec43d86c8a92226339fbd3663ac08416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> (elle donne <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2473754699626fe7b5995a5c28ced9fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#95;&#48;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"\/> sinon). On en d\u00e9duit que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-57d127f0fcb8cba16bc669031fff511b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\"\/>.<\/p>\n<h3>Exercice 2 : passer \u00e0 la suite des inverses<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> D\u00e9terminer la suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> d\u00e9finie par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-256eecca4bc236827fc5d910e583c5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-117249680c98bb1a827fb5c549847f43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/> et, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5f2773b0fe1b56619efc8db20b876ffd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#117;&#95;&#123;&#110;&#45;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"149\" style=\"vertical-align: -36px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Il faut d&rsquo;abord justifier que la suite existe. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f72ed89a1321deea22a0b6b680c7b882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: -5px;\"\/> d\u00e9pend de deux termes, on utilise une r\u00e9currence double avec l&rsquo;hypoth\u00e8se <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d71fa2c45fe455eaaf27ef84a8b6d1ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\"\/> : \u00ab <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-660df71358938adf6c0182b307df64e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9ecfafdba477b0d832d2b94237e6daef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -5px;\"\/> \u00bb.<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a4fc152da9c0802275c766010d183a54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: -3px;\"\/> est vraie puisque <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-256eecca4bc236827fc5d910e583c5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-117249680c98bb1a827fb5c549847f43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/>.<\/li>\n<li>Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d71fa2c45fe455eaaf27ef84a8b6d1ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\"\/> est vraie, alors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9e628a4cbbaddfa905e60ad8c20b0f40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#117;&#95;&#110;&#125;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"116\" style=\"vertical-align: -17px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f72ed89a1321deea22a0b6b680c7b882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: -5px;\"\/> est d\u00e9fini et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a7b14b301d5c9c802aa99e48547cfcb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#50;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -5px;\"\/>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0786d4b404bd2a4a7e3f8690138dd1ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -5px;\"\/> est vraie.<\/li>\n<\/ul>\n<p>La suite est donc d\u00e9finie et \u00e0 valeurs strictement positives. Posons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-755ced7d17e2edcb8ebfd2fa758155e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#117;&#95;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"63\" style=\"vertical-align: -15px;\"\/>. La relation devient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9306a61636aecc2aaebf81b7f8c155a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#118;&#95;&#123;&#110;&#43;&#50;&#125;&#61;&#118;&#95;&#123;&#110;&#43;&#49;&#125;&#43;&#118;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"142\" style=\"vertical-align: -5px;\"\/> : c&rsquo;est une suite r\u00e9currente lin\u00e9aire d&rsquo;ordre 2, d&rsquo;\u00e9quation caract\u00e9ristique <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-252239dd5c7571a63d38e0279d0fc83c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#114;&#94;&#50;&#45;&#114;&#45;&#49;&#61;&#40;&#50;&#114;&#43;&#49;&#41;&#40;&#114;&#45;&#49;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"256\" style=\"vertical-align: -5px;\"\/>. Il existe donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6fb1eb656fef2255bd55204880748a6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#44;&#92;&#109;&#117;&#41;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"83\" style=\"vertical-align: -5px;\"\/> tels que, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9144b149713edbd309202e7b15e98fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#94;&#110;&#43;&#92;&#109;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"149\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>Avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e25e2e1af9e6b37113fc7a4533fa75a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#48;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4dd5680c8e74282efa5abb46b671b683_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#49;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"51\" style=\"vertical-align: -12px;\"\/>, on obtient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-da4033db021b5ae5da4321be1b57f02e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#43;&#92;&#109;&#117;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f097cb7debbc21ca6429a8c5382f65d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#125;&#123;&#50;&#125;&#43;&#92;&#109;&#117;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"94\" style=\"vertical-align: -12px;\"\/>, d&rsquo;o\u00f9 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5d178a9f07060bd796f236319460545c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"45\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-88cef7d7a8ac8e19d2183e88951f16be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#117;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"45\" style=\"vertical-align: -12px;\"\/>. Finalement :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-249e32fa1ecc4ad2d3df4d742972e083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#116;&#105;&#109;&#101;&#115;&#50;&#94;&#110;&#125;&#123;&#50;&#94;&#123;&#110;&#43;&#49;&#125;&#43;&#40;&#45;&#49;&#41;&#94;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"148\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>V\u00e9rification : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3f71d931ce62dcc1ffdd630fd39356b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#51;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"86\" style=\"vertical-align: -12px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7f83c193c9475c63132f164c5a78f3e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#51;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"86\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3d9c572b55ad53d9cc194b0140eb4500_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#50;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#57;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"98\" style=\"vertical-align: -12px;\"\/>, qui est bien la moyenne harmonique de 1 et 2.<\/p>\n<h3>Exercice 3 : passer au logarithme<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-60bad55f77079900f94917f6b15028a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1757855991ceed08ce6584e00f94dfec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> et, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-588616762f1acc1b153991dd6fa4f83c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2ccd540104d1de18aa054b553ba2fc9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#94;&#51;&#92;&#44;&#117;&#95;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#94;&#123;&#49;&#47;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"157\" style=\"vertical-align: -7px;\"\/>. D\u00e9terminer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Une r\u00e9currence double, comme dans l&rsquo;exercice 2, montre que la suite est d\u00e9finie et que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-660df71358938adf6c0182b307df64e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>. On peut donc poser <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b7d5beaaab65a98cd990b7d65cf448ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#92;&#108;&#110;&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"\/>. En prenant le logarithme de la relation :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a06c479917b406df13ae232e2709fc2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#118;&#95;&#123;&#110;&#43;&#50;&#125;&#61;&#51;&#118;&#95;&#123;&#110;&#43;&#49;&#125;&#43;&#118;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"151\" style=\"vertical-align: -5px;\"\/><\/p>\n<p>L&rsquo;\u00e9quation caract\u00e9ristique <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cc70ddef836bde9a9e59f1425c44065b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#114;&#94;&#50;&#45;&#51;&#114;&#45;&#49;&#61;&#40;&#52;&#114;&#43;&#49;&#41;&#40;&#114;&#45;&#49;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"265\" style=\"vertical-align: -5px;\"\/> a pour racines <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-486fe9dd550eb19aae8da7e141a61a4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\"\/>. Il existe <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6fb1eb656fef2255bd55204880748a6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#44;&#92;&#109;&#117;&#41;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"83\" style=\"vertical-align: -5px;\"\/> tels que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8d926147d49b1db328937033f3fa678a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#94;&#110;&#43;&#92;&#109;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"149\" style=\"vertical-align: -17px;\"\/>, avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-224138c2a36212199bb5473c76c907f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#48;&#61;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#43;&#92;&#109;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c0b3cf9e56f9e476f05f6eb76fdb0aef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#49;&#61;&#92;&#109;&#117;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"85\" style=\"vertical-align: -12px;\"\/>. On en tire <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e311cda17c442712cde7b513c09b23d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#109;&#117;&#61;&#118;&#95;&#48;&#43;&#52;&#118;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -4px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7823bb37ee025264e83765942428a547_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#125;&#123;&#52;&#125;&#61;&#118;&#95;&#48;&#45;&#118;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"98\" style=\"vertical-align: -12px;\"\/>, donc :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e2bb4d4bff5ca73a421cf730789d7cce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#40;&#118;&#95;&#48;&#45;&#118;&#95;&#49;&#41;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#45;&#49;&#41;&#94;&#110;&#125;&#123;&#52;&#94;&#123;&#110;&#45;&#49;&#125;&#125;&#43;&#118;&#95;&#48;&#43;&#52;&#118;&#95;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"282\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>En revenant \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c07daf15135c9a859331e391a94089f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#101;&#125;&#94;&#123;&#118;&#95;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"65\" style=\"vertical-align: -3px;\"\/> :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a54bde98e17854a02ba5638b5926c9f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#117;&#95;&#48;&#94;&#123;&#49;&#47;&#53;&#125;&#92;&#44;&#117;&#95;&#49;&#94;&#123;&#52;&#47;&#53;&#125;&#92;&#44;&#92;&#101;&#120;&#112;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#48;&#125;&#123;&#117;&#95;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#45;&#49;&#41;&#94;&#110;&#125;&#123;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#52;&#94;&#123;&#110;&#45;&#49;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"307\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>En particulier, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ec9fb68d2e3fc64f094acf7f136ae1d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#94;&#123;&#49;&#47;&#53;&#125;&#92;&#44;&#117;&#95;&#49;&#94;&#123;&#52;&#47;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"66\" style=\"vertical-align: -5px;\"\/>.<\/p>\n<h2>Comment calculer la limite d&rsquo;une suite simple ?<\/h2>\n<p>Face \u00e0 une forme ind\u00e9termin\u00e9e, trois r\u00e9flexes suffisent dans la plupart des cas : factoriser par le terme dominant, multiplier par la quantit\u00e9 conjugu\u00e9e, ou passer au logarithme quand l&rsquo;exposant d\u00e9pend de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>. Les limites usuelles sont rappel\u00e9es dans le cours sur les <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-limites-et-continuite-maths-sup\/\">limites et la continuit\u00e9<\/a>.<\/p>\n<h3>Exercice 4 : factoriser dans un logarithme<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0b966dc7558149a216706ece036d2fc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#110;&#40;&#110;&#94;&#50;&#43;&#110;&#43;&#49;&#41;&#125;&#123;&#92;&#108;&#110;&#40;&#110;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"155\" style=\"vertical-align: -17px;\"\/>. D\u00e9terminer la limite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> On factorise par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a715fc0f84b8bee4928e92097d5eabfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\"\/> dans le logarithme :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-70cc8b4633397861d04fb33014048849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#110;&#94;&#50;&#43;&#110;&#43;&#49;&#41;&#61;&#50;&#92;&#108;&#110;&#40;&#110;&#41;&#43;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#94;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"336\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f48809cc9dd6fc2fa5a140b637d534d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#50;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#108;&#110;&#40;&#110;&#41;&#125;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#94;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"249\" style=\"vertical-align: -17px;\"\/>. Les deux facteurs du second terme tendent vers 0, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5aefa22be52dfda345452b4836a4efdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#110;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<h3>Exercice 5 : encadrer un sinus<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-97cf51f0fc1e1896d156f13750f57dcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#110;&#92;&#115;&#105;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#110;&#94;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"122\" style=\"vertical-align: -12px;\"\/>. D\u00e9terminer la limite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Montrons que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c3631beb6d75795a2135f79fa00f4dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#92;&#115;&#105;&#110;&#40;&#120;&#41;&#92;&#108;&#101;&#113;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -5px;\"\/> pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-dd1d5933619c93f3f79ea4c294db8847_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#105;&#110;&#091;&#48;&#44;&#92;&#112;&#105;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -5px;\"\/>. La fonction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c57f59fe47798af5606723a13ff4a10e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#58;&#120;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#120;&#45;&#92;&#115;&#105;&#110;&#40;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"\/> a pour d\u00e9riv\u00e9e <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-829a696a866c15124747dc9454ec1904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#99;&#111;&#115;&#40;&#120;&#41;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"\/> : elle est croissante et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-40c53cbe95829c0beaa7f9db5a45786c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#40;&#48;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-571cb5c7fe9bd4a809eedc002de0a7a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\"\/>. Le sinus est positif sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ca409902120b1e6a7482ba6b33fedb74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#091;&#48;&#44;&#92;&#112;&#105;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -5px;\"\/>.<\/p>\n<p>Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9af1c57cb34c5f6ff2559730c27e4be5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#110;&#94;&#50;&#125;&#92;&#105;&#110;&#091;&#48;&#44;&#92;&#112;&#105;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"77\" style=\"vertical-align: -12px;\"\/>, on obtient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-bfa4db97ef4d8550aa398bb8ab0ddf42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#110;&#92;&#116;&#105;&#109;&#101;&#115;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#110;&#94;&#50;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"167\" style=\"vertical-align: -12px;\"\/>. Par encadrement, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-99da9ae044089ab3490ae0b3a9d078f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#110;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"101\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<h3>Exercice 6 : la quantit\u00e9 conjugu\u00e9e<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6b35745b5bb96fd6e04f9a528ea39e1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#94;&#50;&#43;&#110;&#43;&#49;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#94;&#50;&#45;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"257\" style=\"vertical-align: -3px;\"\/>. D\u00e9terminer la limite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> On multiplie et on divise par la quantit\u00e9 conjugu\u00e9e, puis on factorise par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> au d\u00e9nominateur :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-450f8568ad0bf4cd7d87e57a48f39b66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#110;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#94;&#50;&#43;&#110;&#43;&#49;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#94;&#50;&#45;&#110;&#43;&#49;&#125;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#49;&#110;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#94;&#50;&#125;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#49;&#110;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#94;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"511\" style=\"vertical-align: -30px;\"\/><\/p>\n<p>Le d\u00e9nominateur tend vers 2, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-de241a214048b9fd91e3578cabe8f755_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<h3>Exercice 7 : un exposant qui d\u00e9pend de n<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9df5761c476d8b45a444d4186261e895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#40;&#110;&#43;&#49;&#41;&#94;&#123;&#49;&#47;&#92;&#108;&#110;&#40;&#110;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"142\" style=\"vertical-align: -5px;\"\/>. D\u00e9terminer la limite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> On a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-660df71358938adf6c0182b307df64e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f78456435fbe781f4352fdb129ef88f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#117;&#95;&#110;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#110;&#40;&#110;&#43;&#49;&#41;&#125;&#123;&#92;&#108;&#110;&#40;&#110;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"143\" style=\"vertical-align: -17px;\"\/>. En \u00e9crivant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9e408637ded96144f2903d4409bd30a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#110;&#43;&#49;&#41;&#61;&#92;&#108;&#110;&#40;&#110;&#41;&#43;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"240\" style=\"vertical-align: -17px;\"\/> :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-913c43a7f208dae4ae7447fd7cd4888c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#117;&#95;&#110;&#41;&#61;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#108;&#110;&#40;&#110;&#41;&#125;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#111;&#110;&#103;&#114;&#105;&#103;&#104;&#116;&#97;&#114;&#114;&#111;&#119;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"283\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>Par continuit\u00e9 de la <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-fonctions-usuelles-maths-sup\/\">fonction exponentielle<\/a>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7a0e0e2528d455a39451ff2a9af45d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#110;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<h3>Exercice 8 : une croissance compar\u00e9e<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ccb6c488fcc9dc70feebc1e18ee10dca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#40;&#92;&#108;&#110;&#32;&#110;&#41;&#94;&#123;&#49;&#47;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"108\" style=\"vertical-align: -5px;\"\/>. D\u00e9terminer la limite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0709fa4a4481907e0c4e3db86323754b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#32;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\"\/>, donc on peut \u00e9crire :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3861ebc436d1ed68fb4898983da9a8eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#117;&#95;&#110;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#110;&#40;&#92;&#108;&#110;&#32;&#110;&#41;&#125;&#123;&#110;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#110;&#40;&#92;&#108;&#110;&#32;&#110;&#41;&#125;&#123;&#92;&#108;&#110;&#32;&#110;&#125;&#92;&#116;&#105;&#109;&#101;&#115;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#110;&#32;&#110;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"269\" style=\"vertical-align: -12px;\"\/><\/p>\n<p>Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e508f552c8310cc25ac6d4baf0433ac9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#116;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#110;&#32;&#116;&#125;&#123;&#116;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"107\" style=\"vertical-align: -13px;\"\/>, chacun des deux facteurs tend vers 0 (le premier avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2613515d62d816a6fe2482e703ba24e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#108;&#110;&#32;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: 0px;\"\/>). Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-79ad712bb52c1031be35323af166982b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#117;&#95;&#110;&#41;&#92;&#116;&#111;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -5px;\"\/> et, par continuit\u00e9 de l&rsquo;exponentielle, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-de241a214048b9fd91e3578cabe8f755_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<div style=\"background: #FFF8EC; border-left: 6px solid #C77F0C; padding: 15px; margin: 20px 0; color: #0f2742;\"><strong>&#x1f4a1; Astuce :<\/strong> d\u00e8s qu&rsquo;une suite s&rsquo;\u00e9crit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-69814d5346431345a24173cc6df83986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;&#94;&#123;&#98;&#95;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\"\/> avec un exposant qui d\u00e9pend de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>, passez \u00e0 la forme exponentielle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-83a00ba89401786cd73be5a7c1a6e2f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#101;&#125;&#94;&#123;&#98;&#95;&#110;&#92;&#108;&#110;&#40;&#97;&#95;&#110;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: 0px;\"\/>. Vous n&rsquo;avez plus qu&rsquo;une limite de produit \u00e0 \u00e9tudier, puis la continuit\u00e9 de l&rsquo;exponentielle conclut.<\/div>\n<h2>Comment encadrer une suite pour trouver sa limite ?<\/h2>\n<p>Quand la suite est une somme dont on ne conna\u00eet pas la valeur exacte, on encadre chaque terme, puis on somme les in\u00e9galit\u00e9s. Le th\u00e9or\u00e8me d&rsquo;encadrement (dit \u00ab des gendarmes \u00bb) conclut si les deux bornes ont la m\u00eame limite.<\/p>\n<h3>Exercice 9 : une somme de racines (oral Mines-T\u00e9l\u00e9com MP 2018)<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> \u00c9tudier la convergence de la suite de terme g\u00e9n\u00e9ral <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-32137111369fa1e001336716606be230_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#110;&#94;&#50;&#125;&#94;&#123;&#40;&#110;&#43;&#49;&#41;&#94;&#50;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#32;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"116\" style=\"vertical-align: -23px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> La somme compte <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cef4ecab0a1d8984f49aa8dac0abb080_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#110;&#43;&#49;&#41;&#94;&#50;&#45;&#110;&#94;&#50;&#43;&#49;&#61;&#50;&#110;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"205\" style=\"vertical-align: -5px;\"\/> termes. Chacun est compris entre <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d2e787ffcb079c0e0b8f602733146e26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#40;&#110;&#43;&#49;&#41;&#94;&#50;&#125;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"150\" style=\"vertical-align: -20px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2ead3a196b51a6522f3f18e1aee25cd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#94;&#50;&#125;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"72\" style=\"vertical-align: -16px;\"\/>. Donc :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-02681d7697d22a31234fabfbad60ffc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#110;&#43;&#50;&#125;&#123;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#110;&#43;&#50;&#125;&#123;&#110;&#125;&#92;&#113;&#117;&#97;&#100;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#105;&#116;&#125;&#92;&#113;&#117;&#97;&#100;&#50;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#50;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"354\" style=\"vertical-align: -14px;\"\/><\/p>\n<p>Par encadrement, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers 2.<\/p>\n<h3>Exercice 10 : montrer d&rsquo;abord que la suite est born\u00e9e<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5e32a41b55978601a0a4f7f1688d2462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#92;&#105;&#110;&#92;&#44;&#093;&#48;&#44;&#49;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -5px;\"\/> et, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-588616762f1acc1b153991dd6fa4f83c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-44cbb68d76664a8c7310f291b29ebd83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#125;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"135\" style=\"vertical-align: -14px;\"\/>. \u00c9tudier la suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Montrons par r\u00e9currence la propri\u00e9t\u00e9 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d71fa2c45fe455eaaf27ef84a8b6d1ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\"\/> : \u00ab <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-aa842a84c95e5ded749ca339f914b6ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#32;&#117;&#95;&#110;&#60;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"83\" style=\"vertical-align: -3px;\"\/> \u00bb.<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a4fc152da9c0802275c766010d183a54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: -3px;\"\/> est vraie, et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-edfe2955fbee758d4113fd7ccf3cd3cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\"\/> aussi car <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1e2f18cefa405640fecd6245fbf68ad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#49;&#43;&#117;&#95;&#48;&#92;&#105;&#110;&#92;&#44;&#093;&#49;&#44;&#50;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -5px;\"\/>.<\/li>\n<li>Supposons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d71fa2c45fe455eaaf27ef84a8b6d1ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\"\/> vraie pour un <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>. Alors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9ecfafdba477b0d832d2b94237e6daef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -5px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d08de2adf34b440d656d947a0b787d13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#60;&#32;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"168\" style=\"vertical-align: -14px;\"\/> car <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-30e66c0d870dc4da7fdd10ee41e0be21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#43;&#49;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"73\" style=\"vertical-align: -3px;\"\/>.<\/li>\n<\/ul>\n<p>La suite est born\u00e9e, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2c2380427675681eff726b0f19c2de2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#125;&#123;&#110;&#43;&#49;&#125;&#92;&#116;&#111;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"80\" style=\"vertical-align: -14px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ddec045e8ef128d578de32ba18b827e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"118\" style=\"vertical-align: -13px;\"\/>. La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers 1.<\/p>\n<h3>Exercice 11 : trouver une suite auxiliaire monotone<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1757855991ceed08ce6584e00f94dfec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> et, pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-73c096b18522f146bdf3403cd9238c56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#110;&#92;&#44;&#117;&#95;&#110;&#125;&#123;&#110;&#43;&#49;&#43;&#117;&#95;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"145\" style=\"vertical-align: -15px;\"\/>. \u00c9tudier la convergence de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Par r\u00e9currence imm\u00e9diate, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-660df71358938adf6c0182b307df64e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>. On r\u00e9\u00e9crit la relation :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d59b5fb1573b929b07e4d14b40d83bea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#110;&#43;&#49;&#41;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#43;&#117;&#95;&#110;&#92;&#44;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#110;&#92;&#44;&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -5px;\"\/><\/p>\n<p>Avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-280b5f0909678dcf38f604def6a8d3ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#110;&#92;&#44;&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"73\" style=\"vertical-align: -3px;\"\/>, cela donne <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4a56bde973b69a911056b86ad56b67b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#45;&#118;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#117;&#95;&#110;&#92;&#44;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"190\" style=\"vertical-align: -5px;\"\/>. La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> est d\u00e9croissante et positive, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-759335c909d230b4cdea8051d4d18c4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#32;&#118;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#118;&#95;&#49;&#61;&#117;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"130\" style=\"vertical-align: -3px;\"\/>, soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5059ea53fe7fb0142fa25d16fb4fb75a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#32;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#49;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"95\" style=\"vertical-align: -12px;\"\/>. Par encadrement, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-99da9ae044089ab3490ae0b3a9d078f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#117;&#95;&#110;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"101\" style=\"vertical-align: -13px;\"\/>.<\/p>\n\t\t<div data-elementor-type=\"section\" data-elementor-id=\"184597\" class=\"elementor elementor-184597\">\n\t\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-inner-section elementor-element elementor-element-f6a24df elementor-section-content-bottom full_width elementor-section-full_width elementor-section-height-min-height elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"f6a24df\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-narrow\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-66 elementor-inner-column elementor-element elementor-element-1a43b36a ae-bg-gallery-type-default\" data-id=\"1a43b36a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-51eac563 elementor-widget elementor-widget-heading\" data-id=\"51eac563\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">PROF DE MATHS PARTICULIER<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-51d48fa4 elementor-widget elementor-widget-heading\" data-id=\"51d48fa4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Des cours de qualit\u00e9 et enseignants aguerris<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7fd3aeff elementor-widget elementor-widget-heading\" data-id=\"7fd3aeff\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Pr\u00e9parer des concours ou s'exercer<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-691aa52c elementor-align-left elementor-mobile-align-center elementor-widget elementor-widget-button\" data-id=\"691aa52c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-md\" href=\"https:\/\/groupe-reussite.fr\/cours-particuliers\/maths\/tous-niveaux\/france\/\" target=\"_blank\" id=\"CTA_PRD_CP\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Cours de maths \u00e0 domicile<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-266b70e2 elementor-widget elementor-widget-text-editor\" data-id=\"266b70e2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: left\"><b>Avis Google France <\/b><span style=\"color: #e5a430\"><b>\u2605\u2605\u2605\u2605\u2605<\/b><\/span><b> 4,9 sur 5<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-4bbd68df ae-bg-gallery-type-default\" data-id=\"4bbd68df\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n<h2>Comment trouver un \u00e9quivalent d&rsquo;une suite d\u00e9finie par r\u00e9currence ?<\/h2>\n<p>Pour obtenir le comportement pr\u00e9cis d&rsquo;une suite, on proc\u00e8de par \u00e9tapes : une majoration, puis un \u00e9quivalent, puis le terme suivant du d\u00e9veloppement. Chaque question s&rsquo;appuie sur la pr\u00e9c\u00e9dente.<\/p>\n<p><strong>Exercice 12. \u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-621b36cb74b467341f8a84e45eb8fe1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;&#95;&#123;&#110;&#92;&#103;&#101;&#113;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -5px;\"\/> d\u00e9finie par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b7d32691a7aac9636173769518b521c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/> et, pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2a8f386a23585393452b396fa9c28cfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#43;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -5px;\"\/>.<\/p>\n<ol>\n<li>Montrer que pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9fcce9d188009035d021a9fd30b715ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -3px;\"\/>.<\/li>\n<li>D\u00e9terminer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-dc36f2113799bf7cfc981416f5fe0e5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"77\" style=\"vertical-align: -16px;\"\/>.<\/li>\n<li>D\u00e9terminer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-251fd9f330b079ede9bfbc742268b5fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#95;&#110;&#45;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"130\" style=\"vertical-align: -13px;\"\/>.<\/li>\n<\/ol>\n<p><strong>Corrig\u00e9 de la question 1.<\/strong> On raisonne par <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-recurrence-maths-sup\/\">r\u00e9currence<\/a>. Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a51661019cc26a5931f8cb0d5fd63f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-02bd0a4d2935522066b6ca9191a69846_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#49;&#61;&#49;&#60;&#32;&#92;&#115;&#113;&#114;&#116;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -3px;\"\/>. Supposons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9fcce9d188009035d021a9fd30b715ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -3px;\"\/>. Par croissance de la racine carr\u00e9e, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9191b581c6b39acdc8d13e18dd40f660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#43;&#49;&#43;&#117;&#95;&#110;&#125;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#43;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"327\" style=\"vertical-align: -5px;\"\/>. Il reste \u00e0 comparer ce majorant \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fea5f809c71cb859a2f3ddc3b0b639e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -3px;\"\/>. Les r\u00e9els compar\u00e9s \u00e9tant positifs, on peut raisonner par \u00e9quivalence en \u00e9levant au carr\u00e9 :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a9a4ee0b632ced4e79cf02e185a07d4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#43;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#49;&#125;&#125;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#51;&#125;&#92;&#105;&#102;&#102;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#32;&#110;&#43;&#50;&#92;&#105;&#102;&#102;&#50;&#110;&#43;&#49;&#92;&#108;&#101;&#113;&#32;&#110;&#94;&#50;&#43;&#52;&#110;&#43;&#52;&#92;&#105;&#102;&#102;&#32;&#110;&#94;&#50;&#43;&#50;&#110;&#43;&#51;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"582\" style=\"vertical-align: -3px;\"\/><\/p>\n<p>La derni\u00e8re in\u00e9galit\u00e9 est vraie, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1f2cc7646e06b273e7ea7436ba5a1af1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"126\" style=\"vertical-align: -5px;\"\/> et la propri\u00e9t\u00e9 est d\u00e9montr\u00e9e.<\/p>\n<div style=\"background: #FFF8EC; border-left: 6px solid #C77F0C; padding: 15px; margin: 20px 0; color: #0f2742;\"><strong>&#x1f4a1; Astuce :<\/strong> pour deux r\u00e9els positifs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\"\/>, montrer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d43d2ef8a0eafd5ddac0fbc7a2035b64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#113;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -3px;\"\/> revient \u00e0 montrer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1bfd2a27bd8434228ff7a643cd3e3908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#94;&#50;&#92;&#108;&#101;&#113;&#32;&#98;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -3px;\"\/>. Pensez \u00e0 \u00e9crire que les deux membres sont positifs avant d&rsquo;\u00e9lever au carr\u00e9 : c&rsquo;est ce que le correcteur v\u00e9rifie.<\/div>\n<p><strong>Corrig\u00e9 de la question 2.<\/strong> On \u00e9crit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-395e34fbbe595fa815ead12ca0524feb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"141\" style=\"vertical-align: -17px;\"\/>. D&rsquo;apr\u00e8s la question 1, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2db59f83bf9f8d978bb76f6b3a05cb18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;&#123;&#110;&#125;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#45;&#49;&#125;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"165\" style=\"vertical-align: -12px;\"\/>, qui tend vers 0. Par encadrement, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e0de53efa9b7ca9129bf5899a4541655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;&#123;&#110;&#125;&#92;&#116;&#111;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"75\" style=\"vertical-align: -12px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-47529c7a031d56304bf15e1e72133408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"110\" style=\"vertical-align: -16px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 3.<\/strong> On l\u00e8ve l&rsquo;ind\u00e9termination avec la quantit\u00e9 conjugu\u00e9e, en utilisant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-91b08d4fe39ab99440e69a655bd43e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#94;&#50;&#45;&#110;&#61;&#117;&#95;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\"\/> :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e3a004e8989f629b903a6748b6277ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#45;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;&#123;&#117;&#95;&#110;&#43;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#47;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#45;&#49;&#125;&#125;&#123;&#49;&#43;&#117;&#95;&#110;&#47;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#125;&#92;&#116;&#105;&#109;&#101;&#115;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#110;&#45;&#49;&#125;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"375\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>D&rsquo;apr\u00e8s la question 2, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-819b41713b89f969efe3a052bef8bea9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#45;&#49;&#125;&#125;&#92;&#116;&#111;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"93\" style=\"vertical-align: -16px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c88450dc864f7027890e5893c2063d8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#117;&#95;&#110;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#125;&#92;&#116;&#111;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"63\" style=\"vertical-align: -16px;\"\/>, et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f8f66ec21e702ac737191011288cac20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#110;&#45;&#49;&#125;&#123;&#110;&#125;&#125;&#92;&#116;&#111;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"97\" style=\"vertical-align: -15px;\"\/>. Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ad27871844517365fc571dec349bd885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#95;&#110;&#45;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"167\" style=\"vertical-align: -13px;\"\/>. Autrement dit, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b191949b1497ccc9640e8f39394682f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#115;&#113;&#114;&#116;&#32;&#110;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;&#43;&#111;&#40;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"155\" style=\"vertical-align: -12px;\"\/>.<\/p>\n<h2>Une suite born\u00e9e v\u00e9rifiant 2u<sub>n<\/sub> \u2264 u<sub>n-1<\/sub> + u<sub>n+1<\/sub> converge-t-elle ?<\/h2>\n<p><strong>Exercice 13. \u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> une suite born\u00e9e telle que, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a771662c9cdcb735a4865595190856de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#43;&#117;&#95;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"145\" style=\"vertical-align: -5px;\"\/>. On pose <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-753741ad8e0d240a8942ea02273147ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#117;&#95;&#110;&#45;&#117;&#95;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"117\" style=\"vertical-align: -3px;\"\/> pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>.<\/p>\n<ol>\n<li>Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-71d49a67e21ff19bc2d0e88f989e3c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;&#95;&#123;&#110;&#92;&#103;&#101;&#113;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> converge.<\/li>\n<li>Montrer par l&rsquo;absurde que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> converge vers 0.<\/li>\n<li>En d\u00e9duire que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge.<\/li>\n<\/ol>\n<p><strong>Corrig\u00e9 de la question 1.<\/strong> On a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8ae123c45076add63699e3ec1ab5ac13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#118;&#95;&#110;&#61;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#43;&#117;&#95;&#123;&#110;&#45;&#49;&#125;&#45;&#50;&#117;&#95;&#110;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"276\" style=\"vertical-align: -5px;\"\/> : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> est croissante. Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-65cf5ce4e618b170ebab790f126adbb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#95;&#110;&#124;&#92;&#108;&#101;&#113;&#32;&#77;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>, alors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-abec7a356ecf72578b1255b0e390245c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#118;&#95;&#110;&#124;&#92;&#108;&#101;&#113;&#50;&#77;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"\/>. Croissante et major\u00e9e, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> converge vers un r\u00e9el <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 2.<\/strong> Supposons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-04e0209d9e2844928bdbc0e6fe7801ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\"\/>. Il existe <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> tel que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1d8b70ca582e705e5ff3fa4edc034e6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#107;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#32;&#76;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"55\" style=\"vertical-align: -12px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4f60b06133036d042268db835e95df86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#92;&#103;&#101;&#113;&#32;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/>. Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2533dcd02612dbd871f01dcc393a2321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/>, on somme ces in\u00e9galit\u00e9s de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ede1c22ea8c34ee55046b3bb9653fb00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#78;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"79\" style=\"vertical-align: -2px;\"\/> \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1e79ed7f6baca296c1099bda178d9ab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#110;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: -2px;\"\/> : la somme est t\u00e9lescopique et donne<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4ea044ccf2919d84e18fdb2e24d0e1b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#117;&#95;&#78;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#110;&#45;&#78;&#43;&#49;&#41;&#76;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"212\" style=\"vertical-align: -12px;\"\/><\/p>\n<p>Le membre de droite tend vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a043fc7cc21133d8c1d8fdcb8fbbe8eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -2px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e4fea0c10b2b119db78abf03ef60af12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"78\" style=\"vertical-align: -3px;\"\/>, ce qui contredit le fait que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> est born\u00e9e. Le cas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ca3df40f6f4d81ce7c5fe95026c446cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#60;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\"\/> se traite de la m\u00eame fa\u00e7on et conduit \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a7d510140f299235fe0c7cbf8322d9dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#116;&#111;&#45;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"78\" style=\"vertical-align: -3px;\"\/>. Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9bbd75f2b1b94466d0ddb39de533c59f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"45\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 3.<\/strong> La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> est croissante et converge vers 0, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1914f1a1865972850931548addf6510f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#92;&#108;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9a3468467f35bd576bc90810f1625e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"78\" style=\"vertical-align: -3px;\"\/>. La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> est d\u00e9croissante et born\u00e9e : elle converge.<\/p>\n<h2>Comment \u00e9tudier une suite de racines carr\u00e9es embo\u00eet\u00e9es ?<\/h2>\n<p><strong>Exercice 14. \u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-23e84079db2648a7dccdb3d2819c6568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#61;&#92;&#115;&#113;&#114;&#116;&#32;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\"\/> et, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7a2d0459b485eb08dedcda10ba23968a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#117;&#95;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"\/>. On note <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-aece13bceaf794029dca463e8dfc0047_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#43;&#52;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"159\" style=\"vertical-align: -12px;\"\/>.<\/p>\n<ol>\n<li>Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2c7478f0439bf92bd5e9054483baade9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: -3px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>, puis \u00e9tudier la convergence de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>.<\/li>\n<li>Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-67eb10ad7551211e3ceff668301f3a0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#97;&#95;&#110;&#41;&#95;&#123;&#110;&#92;&#103;&#101;&#113;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> une suite telle que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3fb9256c15a5149841b38e82b4dcb781_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#32;&#97;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"84\" style=\"vertical-align: -3px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-06f7dc57b17cfb88bb0f079170fbc9d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#50;&#43;&#92;&#99;&#100;&#111;&#116;&#115;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#110;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"231\" style=\"vertical-align: -10px;\"\/>. Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> converge.<\/li>\n<\/ol>\n<p><strong>Corrig\u00e9 de la question 1.<\/strong> Remarquons d&rsquo;abord que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> est la racine positive de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f379f4351bc84785c00700185a342107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#50;&#45;&#120;&#45;&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"113\" style=\"vertical-align: 0px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cc462066233289964738de8608f549ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#61;&#97;&#43;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"85\" style=\"vertical-align: -2px;\"\/>.<\/p>\n<ul>\n<li><em>Majoration.<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68b4414e11d27f2829a048d90f37bd18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#97;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#32;&#97;&#61;&#117;&#95;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"169\" style=\"vertical-align: -12px;\"\/>. Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2c7478f0439bf92bd5e9054483baade9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: -3px;\"\/>, la croissance de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4a7e88b1186e2828b6d7fa8414370286_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\"\/> donne <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f05d9167307e0b43a56af52a5c5f6f66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#125;&#61;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"210\" style=\"vertical-align: -5px;\"\/>. Par r\u00e9currence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2c7478f0439bf92bd5e9054483baade9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: -3px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<li><em>Monotonie.<\/em> Avec la quantit\u00e9 conjugu\u00e9e, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-588b21ba8bd592729707816388baa29d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#117;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#117;&#95;&#110;&#45;&#117;&#95;&#110;&#94;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#117;&#95;&#110;&#125;&#43;&#117;&#95;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"210\" style=\"vertical-align: -16px;\"\/>. Or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-53f0fe62b2f824f14a4ad6ec898b8f0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#120;&#45;&#120;&#94;&#50;&#61;&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#45;&#120;&#41;&#40;&#120;&#45;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" style=\"vertical-align: -5px;\"\/> avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-da83de2f0006b1fcbd45e6c787ed25d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#43;&#52;&#97;&#125;&#125;&#123;&#50;&#125;&#60;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"165\" style=\"vertical-align: -12px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b5bc5fc89bbd54f021e13bf143b4a260_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"86\" style=\"vertical-align: -3px;\"\/>, les deux facteurs sont positifs et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c28117a73b4f5090e2e414d07e793aac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#92;&#103;&#101;&#113;&#32;&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"79\" style=\"vertical-align: -5px;\"\/>.<\/li>\n<li><em>Limite.<\/em> La suite est croissante et major\u00e9e, elle converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b1806355321442d4733ca02b5d5d2e9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"40\" style=\"vertical-align: -3px;\"\/>. En passant \u00e0 la limite dans <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-27e253ec477092d70e2ebf539ed43288_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#94;&#50;&#61;&#97;&#43;&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"110\" style=\"vertical-align: -7px;\"\/>, on obtient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a764ca454da37691a92ddc3388b02a33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#94;&#50;&#61;&#97;&#43;&#92;&#101;&#108;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -2px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f4291a73c2e75f7ba2a9dc808c79bb4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#92;&#105;&#110;&#92;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6f9b0528cbfce5c4a5894e4128461165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#92;&#103;&#101;&#113;&#48;&#62;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"75\" style=\"vertical-align: -4px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d5d4e7cab0105cb4c039d5825fd48561_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#61;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<\/ul>\n<p><strong>Corrig\u00e9 de la question 2.<\/strong> On a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d0af5dd6c551431d37899330be580a9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#110;&#125;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#110;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#123;&#110;&#43;&#49;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"166\" style=\"vertical-align: -7px;\"\/>. En r\u00e9injectant cette in\u00e9galit\u00e9 de proche en proche dans les racines embo\u00eet\u00e9es, qui sont des fonctions croissantes, on obtient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8ccde4b35fa0238c7171572f7b1e2ae0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#118;&#95;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"75\" style=\"vertical-align: -5px;\"\/>. De plus, en majorant chaque <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-eb74c1c8b6fc6cb7def31be6478a4ac5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"\/> par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0a82d62ef2923a4f246a96208a4690ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"\/> est major\u00e9 par l&rsquo;expression \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> racines <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0af24b763b0c045d081f6158b009b6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#92;&#99;&#100;&#111;&#116;&#115;&#43;&#92;&#115;&#113;&#114;&#116;&#32;&#97;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"164\" style=\"vertical-align: -9px;\"\/>, qui vaut <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4529cbb19f7b2979dc340738a86f3d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"35\" style=\"vertical-align: -3px;\"\/>. Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-54730b5aeca90910aeee9f126510ebdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#95;&#110;&#92;&#108;&#101;&#113;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/>. Croissante et major\u00e9e, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-51da92e7ba3ce84abf08850b5af6dce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#118;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\"\/> converge.<\/p>\n<h2>Comment d\u00e9montrer l&rsquo;existence de la constante d&rsquo;Euler ?<\/h2>\n<p>Cet exercice en six questions est un classique de premi\u00e8re ann\u00e9e. Il relie la s\u00e9rie harmonique au logarithme, puis en d\u00e9duit la somme de la s\u00e9rie harmonique altern\u00e9e. Il pr\u00e9pare directement le chapitre sur les <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-series-numeriques-maths-sup\/\">s\u00e9ries num\u00e9riques<\/a>.<\/p>\n<p><strong>Exercice 15. \u00c9nonc\u00e9.<\/strong> On note <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-dd9312745f93747a670f89d545b2bca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"88\" style=\"vertical-align: -22px;\"\/> pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>.<\/p>\n<ol>\n<li>Montrer que pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f863f699b6e51f250051fa8daed03ef2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-18f686041124269764e2c03eb476505f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#92;&#108;&#110;&#40;&#49;&#43;&#120;&#41;&#92;&#108;&#101;&#113;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"169\" style=\"vertical-align: -14px;\"\/>.<\/li>\n<li>Montrer que la suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0d493a4222dbf18aaa6e8ac88493c6c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;&#61;&#72;&#95;&#110;&#45;&#92;&#108;&#110;&#40;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -5px;\"\/> converge. On note <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/> sa limite.<\/li>\n<li>D\u00e9terminer la limite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d0b262b5abb188d5ebc02e1c4c6780d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#110;&#43;&#49;&#125;&#94;&#123;&#50;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"105\" style=\"vertical-align: -24px;\"\/>.<\/li>\n<li>On note <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-46d7dcfb49d08f588119010b10d01ab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#45;&#49;&#41;&#94;&#123;&#107;&#43;&#49;&#125;&#125;&#123;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"137\" style=\"vertical-align: -22px;\"\/>. Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f04e570386fc562a39cdbc2328bdc690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#83;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"32\" style=\"vertical-align: -5px;\"\/> converge et d\u00e9terminer sa limite.<\/li>\n<li>Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-afe501e5adc20cec98272edbe8ca5822_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#120;&#95;&#110;&#41;&#95;&#123;&#110;&#92;&#103;&#101;&#113;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -5px;\"\/> d\u00e9finie par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4de23ad6e9851a7ec22e3d7e93110c3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#49;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7be9d8dc9153bdf30fdcd5789b62b2e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#123;&#110;&#43;&#49;&#125;&#43;&#120;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"114\" style=\"vertical-align: -12px;\"\/>. Exprimer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2c83758b12d1eb192c053e5f0ac1a434_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> en fonction de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8d9b458367f56fd86c70bcf9fec95dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-01a7b7b5dca66cb33a1207e1f39c1140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\"\/> pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-68e2940245f6b908b8fb44957014be33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, puis donner une condition n\u00e9cessaire et suffisante pour que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8d78fa0de26ba0b8d74f0c83e35d1228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#120;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"31\" style=\"vertical-align: -5px;\"\/> converge.<\/li>\n<li>\u00c9tudier la convergence des suites <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7803cbf06de7bf2f5f243b7e72a35815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#99;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -4px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5721cb262eb2ee521dabbd7b786df9c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#100;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"\/> d\u00e9finies par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e126f2c7de223e3fb320085c42dcb484_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4aa1950e8a12194f56f8e6b9f1f5da84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"15\" style=\"vertical-align: -4px;\"\/> et, pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-edffcee5f00e80940f39138e819ef982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#100;&#95;&#110;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"111\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cf626ac997f535f8cfc33b972ed5a890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#99;&#95;&#110;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"111\" style=\"vertical-align: -12px;\"\/>.<\/li>\n<\/ol>\n<p><strong>Corrig\u00e9 de la question 1.<\/strong> La fonction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a9faed4d187e0586d9a45ff983a56e88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#58;&#120;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#120;&#45;&#92;&#108;&#110;&#40;&#49;&#43;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -5px;\"\/> est <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-derivees-maths-sup\/\">d\u00e9rivable<\/a> sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f9008cbb5ebd262d2ec4491098749391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;&#94;&#43;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\"\/>, avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-dd8a6959dbf9e46d7d52fd2f4dba8e88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#39;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#49;&#125;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"138\" style=\"vertical-align: -14px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-40c53cbe95829c0beaa7f9db5a45786c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#40;&#48;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-571cb5c7fe9bd4a809eedc002de0a7a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\"\/>. De m\u00eame, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-712a053542716f32fa1d454f9cec9171_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#58;&#120;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#92;&#108;&#110;&#40;&#49;&#43;&#120;&#41;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"197\" style=\"vertical-align: -14px;\"\/> v\u00e9rifie <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4e285bbc6626c555789f46369645bd9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#39;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#40;&#120;&#43;&#49;&#41;&#94;&#50;&#125;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"161\" style=\"vertical-align: -17px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-825418c3d29a89d3b7e5e813130bfc2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#40;&#48;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-94486326bbefdf5b2e22feacf8a5c455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 2.<\/strong><\/p>\n<ul>\n<li><em>Monotonie.<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0d8c6018d1467a796b4773ab5baae9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#97;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#49;&#125;&#45;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"253\" style=\"vertical-align: -17px;\"\/>. La question 1 avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8de690bae25b4af81de01032866f03b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"47\" style=\"vertical-align: -12px;\"\/> donne <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-aa9cc8ffa13d3f45b5aabf7b1e61cd56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#92;&#108;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -17px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-477c0e4bc83a875e9b9679481eef3b47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#97;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"\/> est d\u00e9croissante.<\/li>\n<li><em>Minoration.<\/em> La question 1 donne aussi <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-258cf5f015103e510f839e84f59c1736_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#107;&#43;&#49;&#41;&#45;&#92;&#108;&#110;&#40;&#107;&#41;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"164\" style=\"vertical-align: -12px;\"\/>. En sommant de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-40c1016f1b0551bb2411f49448451562_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -1px;\"\/> \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>, la <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-sommes-produits-maths-sup\/\">somme t\u00e9lescopique<\/a> vaut <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-448cd9f0f5d4db8c4d949f2cb9c861cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#40;&#110;&#43;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c1e80b9aa4109a5850af06e3051eabc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;&#95;&#110;&#92;&#103;&#101;&#113;&#92;&#108;&#110;&#40;&#110;&#43;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-91d075f8d16671a389630cd4d63da851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;&#92;&#103;&#101;&#113;&#92;&#108;&#110;&#40;&#110;&#43;&#49;&#41;&#45;&#92;&#108;&#110;&#40;&#110;&#41;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -5px;\"\/>.<\/li>\n<\/ul>\n<p>D\u00e9croissante et minor\u00e9e par 0, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-477c0e4bc83a875e9b9679481eef3b47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#97;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"\/> converge vers un r\u00e9el <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/>, la <strong>constante d&rsquo;Euler<\/strong> : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b02472a44d87d1a1dfc7546757f10e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#48;&#123;&#44;&#125;&#53;&#55;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"75\" style=\"vertical-align: -4px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 3.<\/strong> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-995859d2e5e20adfe47f2a467c44731e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#95;&#110;&#61;&#72;&#95;&#123;&#50;&#110;&#125;&#45;&#72;&#95;&#110;&#61;&#97;&#95;&#123;&#50;&#110;&#125;&#45;&#97;&#95;&#110;&#43;&#92;&#108;&#110;&#40;&#50;&#110;&#41;&#45;&#92;&#108;&#110;&#40;&#110;&#41;&#61;&#97;&#95;&#123;&#50;&#110;&#125;&#45;&#97;&#95;&#110;&#43;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"479\" style=\"vertical-align: -5px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e0ebce0827196c9b3c89588b6d74f3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#50;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"24\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ed392a9187fc44b002c7f34039bceffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"\/> tendent toutes deux vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a7e7f43aee55dd785db1ae493ab6f01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#82;&#95;&#110;&#61;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"122\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 4.<\/strong> Dans <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-402ea0c4aa9901e835d85681ffcfd3e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#50;&#110;&#125;&#45;&#72;&#95;&#123;&#50;&#110;&#125;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#50;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#45;&#49;&#41;&#94;&#123;&#107;&#43;&#49;&#125;&#45;&#49;&#125;&#123;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"53\" width=\"226\" style=\"vertical-align: -22px;\"\/>, les termes d&rsquo;indice impair sont nuls et ceux d&rsquo;indice pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-bfe7165849aa0261e1c854f60a212720_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#50;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\"\/> valent <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3eb001324521917207e49862c34aec11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#50;&#112;&#125;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"83\" style=\"vertical-align: -16px;\"\/>. Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e30e68381de9bbc4ce36105e886b97f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#50;&#110;&#125;&#45;&#72;&#95;&#123;&#50;&#110;&#125;&#61;&#45;&#72;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"140\" style=\"vertical-align: -3px;\"\/>, soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c90010a3336585e1590b55a2971738ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#50;&#110;&#125;&#61;&#82;&#95;&#110;&#92;&#116;&#111;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"126\" style=\"vertical-align: -3px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-911bfd28c8fb38b38beb7a6aad860f77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#50;&#110;&#43;&#49;&#125;&#61;&#83;&#95;&#123;&#50;&#110;&#125;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"170\" style=\"vertical-align: -14px;\"\/>, on a aussi <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d3457cb6e7be1cee04f0a11871d1cf7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#50;&#110;&#43;&#49;&#125;&#92;&#116;&#111;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -5px;\"\/>. Les deux suites extraites des termes pairs et impairs ayant la m\u00eame limite, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d23c8d0c8e321a59dc88c9b362d01c99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#83;&#95;&#110;&#61;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"119\" style=\"vertical-align: -13px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 5.<\/strong> Posons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8dfcef8b2a9eae79d0b8dbe768f93bfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#95;&#110;&#61;&#40;&#45;&#49;&#41;&#94;&#110;&#32;&#120;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -5px;\"\/>. En multipliant la relation par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-756fc620b9b5e399af011b02ca5dcd3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#45;&#49;&#41;&#94;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -4px;\"\/>, on obtient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fcd07f236b0f46d9d7a07219f0fb6f8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#121;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#45;&#49;&#41;&#94;&#123;&#110;&#43;&#49;&#125;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"165\" style=\"vertical-align: -12px;\"\/>. En sommant de 1 \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f30b71e7fcec69d119f30f67cf09c975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\"\/> : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a0fdb1bd6ac4356e2bb2fc2a9f4d486c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#95;&#110;&#45;&#121;&#95;&#49;&#61;&#83;&#95;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -4px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b03c3e05d4cc8f2eceadf15a80b28983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#95;&#49;&#61;&#45;&#120;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"70\" style=\"vertical-align: -4px;\"\/> :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-28f62a42d864ee4dfa2ac3a8f1376c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#110;&#61;&#40;&#45;&#49;&#41;&#94;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#83;&#95;&#123;&#110;&#45;&#49;&#125;&#45;&#120;&#95;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#113;&#117;&#97;&#100;&#40;&#110;&#92;&#103;&#101;&#113;&#50;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"259\" style=\"vertical-align: -5px;\"\/><\/p>\n<p>Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-63cdefe6bd013fbbb19a120d23d8e892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#123;&#110;&#45;&#49;&#125;&#92;&#116;&#111;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -3px;\"\/>, on a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-168d8bf00fb0df04ed7b45e7eebdb17a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#123;&#50;&#110;&#125;&#92;&#116;&#111;&#92;&#108;&#110;&#50;&#45;&#120;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"118\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3aecc79b3d0523ed2708b31b6f15982b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#123;&#50;&#110;&#43;&#49;&#125;&#92;&#116;&#111;&#32;&#120;&#95;&#49;&#45;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -5px;\"\/>. Ces deux limites sont \u00e9gales si et seulement si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5f9830772fad24186b10abb63ed9777f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#49;&#61;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" style=\"vertical-align: -3px;\"\/>. La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8d78fa0de26ba0b8d74f0c83e35d1228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#120;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"31\" style=\"vertical-align: -5px;\"\/> converge donc si et seulement si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5f9830772fad24186b10abb63ed9777f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#49;&#61;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" style=\"vertical-align: -3px;\"\/>, et sa limite est alors 0.<\/p>\n<p><strong>Corrig\u00e9 de la question 6.<\/strong> La somme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3226e06f6a5fb5c2ff847d33c86b724f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#110;&#61;&#99;&#95;&#110;&#43;&#100;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"97\" style=\"vertical-align: -3px;\"\/> v\u00e9rifie <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-377be6f83e12009dd759d2fde1331486_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#115;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"75\" style=\"vertical-align: -5px;\"\/> : elle est constante, \u00e9gale \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d301678125f60e432020e1a55c0dd765_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#95;&#49;&#43;&#100;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/>. La diff\u00e9rence <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-361e46c30a3f753f8952723c6dbdd378_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#110;&#61;&#99;&#95;&#110;&#45;&#100;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\"\/> v\u00e9rifie <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ee172ef80e661d84fcaab081eb1439ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#45;&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#110;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"124\" style=\"vertical-align: -12px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2c8cdcef12549452dbcea2da5573b3bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#110;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"18\" style=\"vertical-align: -12px;\"\/> satisfait la relation de la question 5 avec le premier terme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ab1d7ae0c1f8349ff2c409bbaa4fb774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#99;&#95;&#49;&#45;&#100;&#95;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"54\" style=\"vertical-align: -12px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cfaf2f1a93d5f8c50f7f3ee3d192b98a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#115;&#95;&#110;&#43;&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#110;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"99\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0c7823db4f3882e2771473723b711712_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#115;&#95;&#110;&#45;&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#110;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"101\" style=\"vertical-align: -12px;\"\/>, les suites <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7803cbf06de7bf2f5f243b7e72a35815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#99;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -4px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5721cb262eb2ee521dabbd7b786df9c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#100;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"\/> convergent si et seulement si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-db45cefd9a63c46198c51cb9806bf147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#100;&#101;&#108;&#116;&#97;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"29\" style=\"vertical-align: -5px;\"\/> converge, c&rsquo;est-\u00e0-dire si et seulement si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7ccbcdb4dcb756516d3c5095db8164ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#95;&#49;&#45;&#100;&#95;&#49;&#61;&#50;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"115\" style=\"vertical-align: -3px;\"\/>. Elles ont alors la m\u00eame limite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f4f7be5a6ecc54fdf1c8d95dd3c075bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#99;&#95;&#49;&#43;&#100;&#95;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"54\" style=\"vertical-align: -12px;\"\/>. Dans le cas contraire, elles divergent toutes les deux.<\/p>\n<h2>Comment calculer la limite d&rsquo;un produit avec la trigonom\u00e9trie ?<\/h2>\n<p><strong>Exercice 16. \u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f25551d7b44d9bb61af0107ec15a07e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#92;&#105;&#110;&#091;&#48;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#50;&#125;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -6px;\"\/> et, pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fa09d87d1a9a28e36a77589dad4c8100_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#112;&#114;&#111;&#100;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#92;&#116;&#97;&#110;&#94;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#94;&#107;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"209\" style=\"vertical-align: -22px;\"\/>. Justifier l&rsquo;existence de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-09f0facf06f87e8948449738b9fe86e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\"\/>, montrer que la suite converge et trouver sa limite.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1c8e6e86153791f1e57a1160b2161d50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e4f1557e891c343543d2683914dc4aa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#94;&#107;&#125;&#60;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"90\" style=\"vertical-align: -12px;\"\/>, donc chaque tangente est d\u00e9finie et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-09f0facf06f87e8948449738b9fe86e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\"\/> existe. Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b7b41acc5cb99fb07aaa07b445eb2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-24d2021bc0c123c4b1442f3a4bcbedbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#95;&#110;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -3px;\"\/>. Supposons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b2eb0a6218608e6a32e28203703db20e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#92;&#110;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"39\" style=\"vertical-align: -4px;\"\/>. La formule de <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-trigonometrie-maths-sup\/\">trigonom\u00e9trie<\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5e00bdd142ea706656bc9bbdc24d3c82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#110;&#40;&#50;&#117;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#116;&#97;&#110;&#40;&#117;&#41;&#125;&#123;&#49;&#45;&#92;&#116;&#97;&#110;&#94;&#50;&#40;&#117;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"172\" style=\"vertical-align: -17px;\"\/> donne, pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7bcfcd184e2970cb1f876fcf4356aeb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#92;&#105;&#110;&#92;&#44;&#093;&#48;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#52;&#125;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -6px;\"\/> :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-cd35a6588ed3e3f3dce54b2157ad726b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#116;&#97;&#110;&#94;&#50;&#40;&#117;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#116;&#97;&#110;&#40;&#117;&#41;&#125;&#123;&#92;&#116;&#97;&#110;&#40;&#50;&#117;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"174\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>En l&rsquo;appliquant \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2cf7d49b575d59f0ea0bae659f93599c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#94;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"54\" style=\"vertical-align: -12px;\"\/>, le produit devient t\u00e9lescopique :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a9a94b72e0a79e9fcd8166ae9b3e5061_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#112;&#114;&#111;&#100;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#116;&#97;&#110;&#40;&#116;&#47;&#50;&#94;&#107;&#41;&#125;&#123;&#92;&#116;&#97;&#110;&#40;&#116;&#47;&#50;&#94;&#123;&#107;&#45;&#49;&#125;&#41;&#125;&#61;&#50;&#94;&#110;&#92;&#44;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#97;&#110;&#40;&#116;&#47;&#50;&#94;&#110;&#41;&#125;&#123;&#92;&#116;&#97;&#110;&#40;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"285\" style=\"vertical-align: -22px;\"\/><\/p>\n<p>Enfin <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-14e736e346120a1cdbfc445b5c5c86c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#94;&#110;&#92;&#116;&#97;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#94;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#116;&#92;&#116;&#105;&#109;&#101;&#115;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#105;&#110;&#40;&#116;&#47;&#50;&#94;&#110;&#41;&#125;&#123;&#116;&#47;&#50;&#94;&#110;&#125;&#92;&#116;&#105;&#109;&#101;&#115;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#99;&#111;&#115;&#40;&#116;&#47;&#50;&#94;&#110;&#41;&#125;&#92;&#116;&#111;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"352\" style=\"vertical-align: -17px;\"\/>, car <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e5ae98ce9fc00a140e35a9e8d89f23fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#105;&#110;&#32;&#120;&#125;&#123;&#120;&#125;&#92;&#116;&#111;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"72\" style=\"vertical-align: -12px;\"\/> quand <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2fcd115aa2b0bfc6d20ee9755629c3c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#116;&#111;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\"\/>. Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25c06dce8a371d4e4fd73a291266bb67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#108;&#105;&#109;&#95;&#123;&#110;&#92;&#116;&#111;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#80;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#92;&#116;&#97;&#110;&#40;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"142\" style=\"vertical-align: -17px;\"\/>.<\/p>\n<h2>Comment montrer qu&rsquo;une suite converge vers ln 2 avec une somme de Riemann ?<\/h2>\n<p>La m\u00eame suite peut se traiter de deux fa\u00e7ons selon le moment de l&rsquo;ann\u00e9e : au premier semestre, on montre seulement qu&rsquo;elle converge ; au second, une fois l&rsquo;<a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-integration-maths-sup\/\">int\u00e9gration<\/a> vue, on calcule sa limite.<\/p>\n<p><strong>Exercice 17. \u00c9nonc\u00e9.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3ec63d58ffceac0bfa8f471900a9d423_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#40;&#110;&#43;&#107;&#41;&#40;&#110;&#43;&#107;&#43;&#49;&#41;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"234\" style=\"vertical-align: -22px;\"\/>.<\/p>\n<ol>\n<li>(Premier semestre) Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> est croissante et major\u00e9e, donc convergente.<\/li>\n<li>(Second semestre) Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-328f3d52931acd4f573849ca043337dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<\/ol>\n<p><strong>Corrig\u00e9 de la question 1.<\/strong> Attention \u00e0 remplacer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d72f4e3699652cfc70b8880515893d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\"\/> aux trois endroits : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-87ed0466ad4b99307d162bb4c8afc0dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#40;&#110;&#43;&#49;&#43;&#107;&#41;&#40;&#110;&#43;&#107;&#43;&#50;&#41;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"53\" width=\"282\" style=\"vertical-align: -22px;\"\/>. Avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8be25d98b6ab276b0d6a05ee95cbbbcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#107;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c40cd2a4a696b1de5edff2fa0d9fb8fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#112;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#40;&#110;&#43;&#112;&#41;&#40;&#110;&#43;&#112;&#43;&#49;&#41;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"219\" style=\"vertical-align: -20px;\"\/>, on obtient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-6b0efc3aad6a391855d487ac49c784eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#112;&#61;&#50;&#125;&#94;&#123;&#110;&#43;&#50;&#125;&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#112;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"123\" style=\"vertical-align: -24px;\"\/>, donc :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a25684f31763c25e90a74d04eae451e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#117;&#95;&#110;&#61;&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#110;&#43;&#49;&#41;&#43;&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#110;&#43;&#50;&#41;&#45;&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#49;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#50;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#49;&#125;&#125;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#51;&#125;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#52;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"582\" style=\"vertical-align: -17px;\"\/><\/p>\n<p>Chacune des deux premi\u00e8res fractions de la parenth\u00e8se est strictement sup\u00e9rieure \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a15f10a7b90a4340112aee63aaa19618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#43;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"66\" style=\"vertical-align: -16px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-069aaaa082cdfb77f1cf67ba1f8e7f50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#117;&#95;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -5px;\"\/>. Enfin <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> est la somme de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> termes inf\u00e9rieurs \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b76c8ed9dbf5ea63088d52280326509c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"11\" style=\"vertical-align: -12px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-157ac95afcb2cda1df13da3577ba2bb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\"\/>. Croissante et major\u00e9e, la suite converge.<\/p>\n<p><strong>Corrig\u00e9 de la question 2.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1c8e6e86153791f1e57a1160b2161d50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a7d3ab287d9734114c82af5d54546f1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#43;&#107;&#92;&#108;&#101;&#113;&#92;&#115;&#113;&#114;&#116;&#123;&#40;&#110;&#43;&#107;&#41;&#40;&#110;&#43;&#107;&#43;&#49;&#41;&#125;&#92;&#108;&#101;&#113;&#32;&#110;&#43;&#107;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"321\" style=\"vertical-align: -6px;\"\/>, donc en sommant :<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c4ba35ee01c80275eb1311292575b954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#107;&#43;&#49;&#125;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#110;&#92;&#108;&#101;&#113;&#32;&#98;&#95;&#110;&#61;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"328\" style=\"vertical-align: -22px;\"\/><\/p>\n<p>On \u00e9crit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-19759c8c0ae94995228427ba868c872f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#95;&#110;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#110;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#115;&#117;&#109;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#32;&#107;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"138\" style=\"vertical-align: -22px;\"\/> avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-369f3e8ce66a1ae4576f8a6e4c487078_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#43;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"101\" style=\"vertical-align: -14px;\"\/>, continue sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25b6d943ab489c05a3dbd5ea29087a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#091;&#48;&#44;&#49;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -5px;\"\/>. C&rsquo;est une somme de Riemann, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4603778ea47790cf6f16b866e4f55762_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#95;&#110;&#92;&#116;&#111;&#92;&#100;&#105;&#115;&#112;&#108;&#97;&#121;&#115;&#116;&#121;&#108;&#101;&#92;&#105;&#110;&#116;&#95;&#48;&#94;&#49;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#116;&#125;&#123;&#49;&#43;&#116;&#125;&#61;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"163\" style=\"vertical-align: -16px;\"\/>. Par ailleurs, avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8be25d98b6ab276b0d6a05ee95cbbbcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#107;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4ee7c64ba0a09364339b77f704491b61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;&#61;&#98;&#95;&#110;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#110;&#43;&#49;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"200\" style=\"vertical-align: -14px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1fa1873fa45118a11dfb6c7c8b4d5c4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#110;&#92;&#116;&#111;&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"72\" style=\"vertical-align: -3px;\"\/> aussi. Par encadrement, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-328f3d52931acd4f573849ca043337dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<h2>Comment \u00e9tudier une suite r\u00e9currente u<sub>n+1<\/sub> = f(u<sub>n<\/sub>) ?<\/h2>\n<p>La m\u00e9thode suit toujours le m\u00eame plan : \u00e9tudier les variations de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/> et le signe de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-22822e890a29961b079db4340fe1e442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"\/>, chercher les points fixes, trouver un intervalle stable par <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/>, en d\u00e9duire la monotonie, puis conclure. La seule limite finie possible d&rsquo;une telle suite est un point fixe de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/>, si <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/> est continue.<\/p>\n<div style=\"background: #FFF8EC; border-left: 6px solid #C77F0C; padding: 15px; margin: 20px 0; color: #0f2742;\"><strong>&#x1f4a1; Astuce :<\/strong> sur un intervalle stable <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>, le sens de variation de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> se lit sur le signe de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-22822e890a29961b079db4340fe1e442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"\/> : si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3aad8a26d83061639975b205ad0c3044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#92;&#103;&#101;&#113;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -5px;\"\/> sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>, alors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2a402d8dd7b66a1f9c40c85a2d05d362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#102;&#40;&#117;&#95;&#110;&#41;&#92;&#103;&#101;&#113;&#32;&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -5px;\"\/> et la suite est croissante. Ce crit\u00e8re \u00e9vite une r\u00e9currence sur la monotonie.<\/div>\n<h3>Exercice 18 : une fonction croissante et deux points fixes<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0b55351a084e8d7959aeabd8ae8f946a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#43;&#50;&#120;&#94;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"118\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> d\u00e9finie par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ec43d86c8a92226339fbd3663ac08416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> et <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-280c9d22efa1f941d0ba972441a4364d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#102;&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"\/>. D\u00e9terminer les suites constantes, puis \u00e9tudier la suite selon la valeur de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fce4ef575cb9dacb190c93ae8d286cdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/> est croissante sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f9008cbb5ebd262d2ec4491098749391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;&#94;&#43;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-07f5edb792e8dee5ec21bc77c944d6af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#45;&#120;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#94;&#50;&#45;&#53;&#120;&#43;&#51;&#125;&#123;&#53;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#50;&#120;&#45;&#51;&#41;&#40;&#120;&#45;&#49;&#41;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"335\" style=\"vertical-align: -12px;\"\/>. Les points fixes sont <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-7a79ab9ddedce88b87defe8a584246f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\"\/> : il y a deux suites constantes, obtenues pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-256eecca4bc236827fc5d910e583c5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-531c368c8f6cbd55d10aef69a7cd3b60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"52\" style=\"vertical-align: -12px;\"\/>. Le signe de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-22822e890a29961b079db4340fe1e442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"\/> est positif sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-4d1009028ab64be275b5d2d784446b5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#091;&#48;&#44;&#49;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -5px;\"\/> et sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-44319287bb3a0abfc30254ff96833c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#093;&#92;&#102;&#114;&#97;&#99;&#51;&#50;&#44;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\"\/>, n\u00e9gatif sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d556080cd666a4601910cf89106023c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#093;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#51;&#50;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"37\" style=\"vertical-align: -6px;\"\/>.<\/p>\n<ul>\n<li><em>Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0ad3d54a5223417c8581081939257caa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#113;&#32;&#117;&#95;&#48;&#60;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\"\/>.<\/em> Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9239441a147d3be3505494cc3bf8d98a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#48;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-585b61e1c7da89bf1a5b44ef089b6267_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#49;&#41;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"\/>, l&rsquo;intervalle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25b6d943ab489c05a3dbd5ea29087a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#091;&#48;&#44;&#49;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -5px;\"\/> est stable par <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/>, et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-3aad8a26d83061639975b205ad0c3044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#92;&#103;&#101;&#113;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -5px;\"\/> sur cet intervalle. La suite est croissante et major\u00e9e par 1, elle converge vers un point fixe de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25b6d943ab489c05a3dbd5ea29087a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#091;&#48;&#44;&#49;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -5px;\"\/>, donc vers 1.<\/li>\n<li><em>Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0a5eac060bba3d71ec88166e00e8b39f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#60;&#32;&#117;&#95;&#48;&#60;&#32;&#92;&#102;&#114;&#97;&#99;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\"\/>.<\/em> L&rsquo;intervalle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-d556080cd666a4601910cf89106023c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#093;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#51;&#50;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"37\" style=\"vertical-align: -6px;\"\/> est stable et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-814ec7e58f1c92fc4be1cd673c5afd73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#60;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -5px;\"\/> : la suite est d\u00e9croissante et minor\u00e9e par 1. Sa limite est un point fixe inf\u00e9rieur ou \u00e9gal \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c245542e1bba65d2bd076e63f9be1bf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#60;&#32;&#92;&#102;&#114;&#97;&#99;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\"\/>, donc la suite converge vers 1.<\/li>\n<li><em>Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e6a01800d7d8fe2e4c6c1019ac17014d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#62;&#92;&#102;&#114;&#97;&#99;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\"\/>.<\/em> L&rsquo;intervalle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-44319287bb3a0abfc30254ff96833c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#093;&#92;&#102;&#114;&#97;&#99;&#51;&#50;&#44;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\"\/> est stable et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fa1c2f7b9d172ef3f3858325274bc798_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#62;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -5px;\"\/> : la suite est croissante. Si elle convergeait, sa limite serait un point fixe sup\u00e9rieur ou \u00e9gal \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e6a01800d7d8fe2e4c6c1019ac17014d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#62;&#92;&#102;&#114;&#97;&#99;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\"\/> ; il n&rsquo;en existe pas. La suite diverge donc vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-a043fc7cc21133d8c1d8fdcb8fbbe8eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -2px;\"\/>.<\/li>\n<\/ul>\n<h3>Exercice 19 : utiliser l&rsquo;in\u00e9galit\u00e9 des accroissements finis<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-dc883ad75f98805dea6fbcabdfadd67a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#32;&#120;&#52;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"115\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> d\u00e9finie par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-60bad55f77079900f94917f6b15028a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#48;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> et <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-280c9d22efa1f941d0ba972441a4364d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#61;&#102;&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"\/>.<\/p>\n<ol>\n<li>Montrer que la suite est bien d\u00e9finie et que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-205e170519d52bebb3108081e6039532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#54;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"69\" style=\"vertical-align: -12px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>.<\/li>\n<li>Quelle est la seule limite possible de la suite ?<\/li>\n<li>Montrer que pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8db18e533376bfc4b706899fea6f5a37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#124;&#117;&#95;&#123;&#110;&#43;&#49;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#124;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#52;&#92;&#108;&#101;&#102;&#116;&#124;&#117;&#95;&#110;&#45;&#92;&#115;&#113;&#114;&#116;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"206\" style=\"vertical-align: -12px;\"\/> et conclure.<\/li>\n<\/ol>\n<p><strong>Corrig\u00e9 de la question 1.<\/strong> Pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-37665e9aef746e6afdd95539e8c09bd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"43\" style=\"vertical-align: -2px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-74d71ad296fdd2fd9e4d5329a42bb98c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#54;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"143\" style=\"vertical-align: -17px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-62ec895dfab95a7393e7e153eb3f7851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#54;&#120;&#92;&#103;&#101;&#113;&#50;&#92;&#115;&#113;&#114;&#116;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"102\" style=\"vertical-align: -12px;\"\/>, car <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-23566c357cb4a7cd7c06926f27657336_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#32;&#120;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#47;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#94;&#50;&#92;&#103;&#101;&#113;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"151\" style=\"vertical-align: -11px;\"\/>. Donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-50b01e40f36c0257430ff6139984ff3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#54;&#125;&#123;&#50;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"118\" style=\"vertical-align: -12px;\"\/>. Par r\u00e9currence, chaque <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> est strictement positif, la suite est bien d\u00e9finie, et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-205e170519d52bebb3108081e6039532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#54;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"69\" style=\"vertical-align: -12px;\"\/> d\u00e8s <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 2.<\/strong> Si la suite converge, sa limite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-60a459ae9865dcfc0c8e6ef5a9a10d2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"7\" style=\"vertical-align: -1px;\"\/> v\u00e9rifie <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b96bb92fbe130f31057f4372ddd112dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#92;&#103;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#54;&#125;&#123;&#50;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"91\" style=\"vertical-align: -12px;\"\/> et, par continuit\u00e9 de <img decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2355d97b845255925ae8c721bb95387c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#61;&#102;&#40;&#92;&#101;&#108;&#108;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -5px;\"\/>, soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-ae088a7ea12f88051a64c0761c5a4f6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#101;&#108;&#108;&#125;&#123;&#52;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#92;&#101;&#108;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"61\" style=\"vertical-align: -12px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5d9457acb73a4338035a2e58c467116d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#94;&#50;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: 0px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-f5787e55390611211f29e0900418736e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#61;&#92;&#115;&#113;&#114;&#116;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9 de la question 3.<\/strong> Sur <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-e03806b584101b67b816b5fe79d04e21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#92;&#108;&#101;&#102;&#116;&#091;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#54;&#125;&#123;&#50;&#125;&#44;&#43;&#92;&#105;&#110;&#102;&#116;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#091;\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"122\" style=\"vertical-align: -23px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1155f9218249a6b2874864c3dc618847_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#39;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#52;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#120;&#94;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"126\" style=\"vertical-align: -12px;\"\/> avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-0db588a516840328df80c6e7c2b9fd2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#120;&#94;&#50;&#125;&#92;&#108;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"95\" style=\"vertical-align: -12px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b7163905c988d685a6cc024ac3a0ed0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#52;&#92;&#108;&#101;&#113;&#32;&#102;&#39;&#40;&#120;&#41;&#60;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"123\" style=\"vertical-align: -12px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-c22a81472d6acc5f004f03fb008cf9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#102;&#39;&#40;&#120;&#41;&#124;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"81\" style=\"vertical-align: -12px;\"\/>. Comme <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-57d127f0fcb8cba16bc669031fff511b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\"\/> appartiennent \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> pour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-608107a752b5ecc8950b495899a01bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\"\/>, l&rsquo;in\u00e9galit\u00e9 des accroissements finis donne <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-fb39195d5cae72dea05c3e219cc411dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#102;&#40;&#117;&#95;&#110;&#41;&#45;&#102;&#40;&#92;&#115;&#113;&#114;&#116;&#50;&#41;&#124;&#92;&#108;&#101;&#113;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#52;&#124;&#117;&#95;&#110;&#45;&#92;&#115;&#113;&#114;&#116;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"230\" style=\"vertical-align: -12px;\"\/>, c&rsquo;est-\u00e0-dire l&rsquo;in\u00e9galit\u00e9 demand\u00e9e puisque <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-8214b5256517c4dded9ff8368ab3c81b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#92;&#115;&#113;&#114;&#116;&#50;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"95\" style=\"vertical-align: -5px;\"\/>. Par r\u00e9currence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-dc4032f41bc09be3dcd8d09bf5beb9e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#95;&#110;&#45;&#92;&#115;&#113;&#114;&#116;&#50;&#124;&#92;&#108;&#101;&#113;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#51;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#94;&#123;&#110;&#45;&#49;&#125;&#124;&#117;&#95;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"236\" style=\"vertical-align: -17px;\"\/>, qui tend vers 0. La suite converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-57d127f0fcb8cba16bc669031fff511b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\"\/>.<\/p>\n\t\t<div data-elementor-type=\"section\" data-elementor-id=\"206003\" class=\"elementor elementor-206003\">\n\t\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t<section class=\"has_ae_slider elementor-section elementor-inner-section elementor-element elementor-element-46248836 elementor-section-content-bottom full_width elementor-section-full_width elementor-section-height-min-height elementor-section-height-default ae-bg-gallery-type-default\" data-id=\"46248836\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;gradient&quot;}\">\n\t\t\t\t\t\t\t<div class=\"elementor-background-overlay\"><\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-narrow\">\n\t\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-56fbe2a1 ae-bg-gallery-type-default\" data-id=\"56fbe2a1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3d9d711c elementor-widget elementor-widget-heading\" data-id=\"3d9d711c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">STAGE MATHS VACANCES<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-66976626 elementor-widget elementor-widget-heading\" data-id=\"66976626\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Se sentir mieux et viser haut en maths !<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-12080dd5 elementor-align-left elementor-mobile-align-center elementor-widget elementor-widget-button\" data-id=\"12080dd5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-md\" href=\"https:\/\/groupe-reussite.fr\/stages\/maths\/tous-niveaux\/france\/toutes-periodes\/\" target=\"_blank\" id=\"CTA_PRD_STAGE\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Stage de maths<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"has_ae_slider elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-382b6f23 ae-bg-gallery-type-default\" data-id=\"382b6f23\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n<h2>Exercices th\u00e9oriques : suites extraites et suites stationnaires<\/h2>\n<p>Ces deux exercices portent sur la d\u00e9finition de la limite et sur le <strong>th\u00e9or\u00e8me de Bolzano-Weierstrass<\/strong> (de toute suite r\u00e9elle born\u00e9e, on peut extraire une suite convergente). Ils tombent souvent en question de cours d\u00e9velopp\u00e9e lors des colles.<\/p>\n<h3>Exercice 20 : une suite born\u00e9e dont toutes les suites extraites convergentes ont la m\u00eame limite<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Soit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> une suite r\u00e9elle born\u00e9e et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-882e30c71b1c8839ce30703d8b5b9ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\"\/>. On suppose que toutes les suites extraites convergentes de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> convergent vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>. Montrer que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Raisonnons par l&rsquo;absurde. Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> ne converge pas vers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>, il existe <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-325a49e7902d2175e373012b9bcf5996_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#118;&#97;&#114;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"\/> tel que, pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/>, il existe <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2533dcd02612dbd871f01dcc393a2321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/> avec <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-edd2bb8ab5f43f5ad76627728b7cf70e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#95;&#110;&#45;&#76;&#124;&#92;&#103;&#101;&#113;&#92;&#118;&#97;&#114;&#101;&#112;&#115;&#105;&#108;&#111;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"\/>. On construit ainsi une application strictement croissante <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-275e3ff85c541772575a0f466b91d2c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#118;&#97;&#114;&#112;&#104;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"\/> telle que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-eec81a0042e6d2ad287da84bc4761e84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#95;&#123;&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#110;&#41;&#125;&#45;&#76;&#124;&#92;&#103;&#101;&#113;&#92;&#118;&#97;&#114;&#101;&#112;&#115;&#105;&#108;&#111;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -8px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>. La suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1a6a22b9dd206bdde5c6b1b6549fea31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#123;&#92;&#118;&#97;&#114;&#112;&#104;&#105;&#40;&#110;&#41;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -8px;\"\/> est born\u00e9e : d&rsquo;apr\u00e8s le th\u00e9or\u00e8me de Bolzano-Weierstrass, elle admet une suite extraite convergente, de limite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-60a459ae9865dcfc0c8e6ef5a9a10d2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"7\" style=\"vertical-align: -1px;\"\/>. C&rsquo;est aussi une suite extraite de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/>, donc <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-9fd78d9e6adb14fa6c018ab0de4582db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#61;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"\/> par hypoth\u00e8se. Mais en passant \u00e0 la limite dans l&rsquo;in\u00e9galit\u00e9, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-024b13e4266947467bec2b1b2885a910_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#101;&#108;&#108;&#45;&#76;&#124;&#92;&#103;&#101;&#113;&#92;&#118;&#97;&#114;&#101;&#112;&#115;&#105;&#108;&#111;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"\/> : contradiction.<\/p>\n<p>L&rsquo;hypoth\u00e8se \u00ab born\u00e9e \u00bb est indispensable. La suite d\u00e9finie par <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2d8273382cdb849edbb33c0f78f3dfce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#50;&#110;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\"\/> et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03a00fb2e01e817dbbf55135dd958aea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#123;&#50;&#110;&#43;&#49;&#125;&#61;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"78\" style=\"vertical-align: -5px;\"\/> diverge, alors que toutes ses suites extraites convergentes tendent vers 0.<\/p>\n<h3>Exercice 21 : une suite convergente qui prend un nombre fini de valeurs<\/h3>\n<p><strong>\u00c9nonc\u00e9.<\/strong> Montrer que si la suite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-182578a146e80cc620a67c5aba685c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#117;&#95;&#110;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\"\/> converge et ne prend qu&rsquo;un nombre fini de valeurs, elle est stationnaire.<\/p>\n<p><strong>Corrig\u00e9.<\/strong> Notons <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/> la limite et <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"\/> l&rsquo;ensemble fini des valeurs prises par la suite. Si <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"\/> ne contient pas d&rsquo;autre valeur que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>, la suite est constante. Sinon, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-1c4048d77a2e9f63b39a64fc679fee4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#101;&#108;&#116;&#97;&#61;&#92;&#109;&#105;&#110;&#92;&#123;&#124;&#97;&#45;&#76;&#124;&#32;&#58;&#32;&#97;&#92;&#105;&#110;&#32;&#65;&#44;&#92;&#32;&#97;&#92;&#110;&#101;&#113;&#32;&#76;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"249\" style=\"vertical-align: -5px;\"\/> est strictement positif, car c&rsquo;est le minimum d&rsquo;un ensemble fini de r\u00e9els strictement positifs. Par d\u00e9finition de la limite, il existe <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> tel que <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5dd9fd49b87e50639eb474066fba2fc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#95;&#110;&#45;&#76;&#124;&#60;&#32;&#92;&#100;&#101;&#108;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"\/> pour tout <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-2533dcd02612dbd871f01dcc393a2321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#113;&#32;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"\/>. Pour ces indices, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-03456a12d9cd528a7496bd99ec6a5e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"\/> est une valeur de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"\/> \u00e0 distance de <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/> strictement inf\u00e9rieure \u00e0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-02c416d77f6650e9c7849397bf6e11bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#101;&#108;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> : c&rsquo;est forc\u00e9ment <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>. La suite est donc constante \u00e0 partir du rang <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/groupe-reussite.fr\/ressources\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<p>Si un exercice vous r\u00e9siste ou si vous voulez reprendre tout le chapitre, un professeur peut vous accompagner en <a href=\"https:\/\/groupe-reussite.fr\/cours-particuliers\/maths\/tous-niveaux\/france\/\">cours particuliers de maths<\/a>, \u00e0 domicile ou en ligne. Groupe R\u00e9ussite organise aussi des <a href=\"https:\/\/groupe-reussite.fr\/stages\/toutes-matieres\/maths-sup\/france\/toutes-periodes\/\">stages intensifs de maths sup<\/a> pendant les vacances, en petits groupes. Pour revoir les d\u00e9finitions et les th\u00e9or\u00e8mes utilis\u00e9s ici, reprenez le <a href=\"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-suites-numeriques-maths-sup\/\">r\u00e9sum\u00e9 de cours sur les suites num\u00e9riques en maths sup<\/a>.<\/p>\n<h2>FAQ sur les suites num\u00e9riques en maths sup<\/h2>\n        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Comment \u00e9tudier une suite r\u00e9currente du type u(n+1) = f(u(n)) en maths sup ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tOn \u00e9tudie d'abord la fonction f : ses variations, le signe de f(x) - x et ses points fixes. On cherche ensuite un intervalle stable par f qui contient le premier terme. Sur cet intervalle, le signe de f(x) - x donne le sens de variation de la suite. Si la suite est monotone et born\u00e9e, elle converge, et sa limite est un point fixe de f quand f est continue. Sinon, l'in\u00e9galit\u00e9 des accroissements finis permet souvent de montrer la convergence vers le point fixe.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Comment trouver le terme g\u00e9n\u00e9ral d&#039;une suite r\u00e9currente lin\u00e9aire d&#039;ordre 2 ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tPour une suite v\u00e9rifiant u(n+2) = a u(n+1) + b u(n), on r\u00e9sout l'\u00e9quation caract\u00e9ristique r\u00b2 = a r + b. Avec deux racines r\u00e9elles distinctes r1 et r2, le terme g\u00e9n\u00e9ral s'\u00e9crit \u03bb r1^n + \u03bc r2^n. Avec une racine double r, il s'\u00e9crit (\u03bb + \u03bc n) r^n. Les constantes \u03bb et \u03bc se calculent avec les deux premiers termes. Avec des racines complexes conjugu\u00e9es, on obtient une combinaison de cosinus et de sinus.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Qu&#039;est-ce que la constante d&#039;Euler ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tLa constante d'Euler, not\u00e9e \u03b3, est la limite de la suite 1 + 1\/2 + ... + 1\/n - ln(n). Elle vaut environ 0,577. Son existence se d\u00e9montre en maths sup en montrant que cette suite est d\u00e9croissante et minor\u00e9e par 0, gr\u00e2ce \u00e0 l'encadrement x\/(x+1) \u2264 ln(1+x) \u2264 x. Elle sert notamment \u00e0 prouver que la s\u00e9rie harmonique altern\u00e9e a pour somme ln(2).                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Quelle diff\u00e9rence entre les suites en terminale et en maths sup ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tEn terminale, on \u00e9tudie surtout les suites arithm\u00e9tiques et g\u00e9om\u00e9triques, le raisonnement par r\u00e9currence et les th\u00e9or\u00e8mes de comparaison. En maths sup, la d\u00e9finition de la limite est formalis\u00e9e avec des epsilon, et le programme ajoute les suites extraites, le th\u00e9or\u00e8me de Bolzano-Weierstrass, les suites adjacentes, les suites r\u00e9currentes lin\u00e9aires d'ordre 2 et l'\u00e9tude syst\u00e9matique des suites u(n+1) = f(u(n)).                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t        <section class=\"sc_fs_faq sc_card \">\n            <div>\n\t\t\t\t<h3>Quand \u00e9tudie-t-on les suites num\u00e9riques en MPSI, PCSI et PTSI ?<\/h3>                <div>\n\t\t\t\t\t                    <p>\n\t\t\t\t\t\tLe chapitre Nombres r\u00e9els et suites num\u00e9riques figure au premier semestre du programme de math\u00e9matiques de premi\u00e8re ann\u00e9e. Les suites reviennent ensuite tout au long de l'ann\u00e9e : analyse asymptotique, sommes de Riemann dans le chapitre d'int\u00e9gration, puis s\u00e9ries num\u00e9riques au second semestre.                    <\/p>\n                <\/div>\n            <\/div>\n        <\/section>\n\t\t\n<script type=\"application\/ld+json\">\n    {\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Comment \u00e9tudier une suite r\u00e9currente du type u(n+1) = f(u(n)) en maths sup ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"On \u00e9tudie d'abord la fonction f : ses variations, le signe de f(x) - x et ses points fixes. On cherche ensuite un intervalle stable par f qui contient le premier terme. Sur cet intervalle, le signe de f(x) - x donne le sens de variation de la suite. Si la suite est monotone et born\u00e9e, elle converge, et sa limite est un point fixe de f quand f est continue. Sinon, l'in\u00e9galit\u00e9 des accroissements finis permet souvent de montrer la convergence vers le point fixe.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Comment trouver le terme g\u00e9n\u00e9ral d'une suite r\u00e9currente lin\u00e9aire d'ordre 2 ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Pour une suite v\u00e9rifiant u(n+2) = a u(n+1) + b u(n), on r\u00e9sout l'\u00e9quation caract\u00e9ristique r\u00b2 = a r + b. Avec deux racines r\u00e9elles distinctes r1 et r2, le terme g\u00e9n\u00e9ral s'\u00e9crit \u03bb r1^n + \u03bc r2^n. Avec une racine double r, il s'\u00e9crit (\u03bb + \u03bc n) r^n. Les constantes \u03bb et \u03bc se calculent avec les deux premiers termes. Avec des racines complexes conjugu\u00e9es, on obtient une combinaison de cosinus et de sinus.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Qu'est-ce que la constante d'Euler ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"La constante d'Euler, not\u00e9e \u03b3, est la limite de la suite 1 + 1\/2 + ... + 1\/n - ln(n). Elle vaut environ 0,577. Son existence se d\u00e9montre en maths sup en montrant que cette suite est d\u00e9croissante et minor\u00e9e par 0, gr\u00e2ce \u00e0 l'encadrement x\/(x+1) \u2264 ln(1+x) \u2264 x. Elle sert notamment \u00e0 prouver que la s\u00e9rie harmonique altern\u00e9e a pour somme ln(2).\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Quelle diff\u00e9rence entre les suites en terminale et en maths sup ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"En terminale, on \u00e9tudie surtout les suites arithm\u00e9tiques et g\u00e9om\u00e9triques, le raisonnement par r\u00e9currence et les th\u00e9or\u00e8mes de comparaison. En maths sup, la d\u00e9finition de la limite est formalis\u00e9e avec des epsilon, et le programme ajoute les suites extraites, le th\u00e9or\u00e8me de Bolzano-Weierstrass, les suites adjacentes, les suites r\u00e9currentes lin\u00e9aires d'ordre 2 et l'\u00e9tude syst\u00e9matique des suites u(n+1) = f(u(n)).\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Quand \u00e9tudie-t-on les suites num\u00e9riques en MPSI, PCSI et PTSI ?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Le chapitre Nombres r\u00e9els et suites num\u00e9riques figure au premier semestre du programme de math\u00e9matiques de premi\u00e8re ann\u00e9e. Les suites reviennent ensuite tout au long de l'ann\u00e9e : analyse asymptotique, sommes de Riemann dans le chapitre d'int\u00e9gration, puis s\u00e9ries num\u00e9riques au second semestre.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t    ]\n}\n<\/script>\n\n<h2>Sources<\/h2>\n<ul>\n<li>Minist\u00e8re de l&rsquo;Enseignement sup\u00e9rieur et de la Recherche, <a href=\"https:\/\/www.education.gouv.fr\/bo\/21\/Special1\/ESRS2035779A.htm\">arr\u00eat\u00e9 du 5 janvier 2021, programme de math\u00e9matiques de MP2I et MPSI<\/a>, Bulletin officiel sp\u00e9cial n\u00b0 1 du 11 f\u00e9vrier 2021 (applicable en premi\u00e8re ann\u00e9e depuis la rentr\u00e9e 2021).<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Exercices corrig\u00e9s sur les suites num\u00e9riques en maths sup Par Seda Biber, le 27\/10\/2026 Pour \u00e9tudier une suite num\u00e9rique en maths sup, commencez toujours par reconna\u00eetre sa forme. Une suite arithm\u00e9tico-g\u00e9om\u00e9trique ou r\u00e9currente lin\u00e9aire d&rsquo;ordre 2 se calcule explicitement. Une suite d\u00e9finie par s&rsquo;\u00e9tudie avec les variations de , le signe de et les intervalles [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ocean_post_layout":"full-screen","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"ressource","ocean_second_sidebar":"0","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"0","ocean_custom_header_template":"0","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"0","ocean_menu_typo_font_family":"0","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"on","ocean_disable_heading":"on","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"on","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"0","footnotes":""},"class_list":["post-37456","page","type-page","status-publish","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - 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