{"id":234971,"date":"2022-06-28T17:18:07","date_gmt":"2022-06-28T15:18:07","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=234971"},"modified":"2025-09-29T11:48:23","modified_gmt":"2025-09-29T09:48:23","slug":"exemple-des-suites-geometriques","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/exemple-des-suites-geometriques\/","title":{"rendered":"Exemple des suites g\u00e9om\u00e9triques"},"content":{"rendered":"\n

Vous trouvez les suites g\u00e9om\u00e9triques<\/a> complexes ? Un cours particulier de maths<\/strong><\/a> avec nos profs peut vous aider \u00e0 d\u00e9mystifier leurs secrets. \ud83c\udf00<\/p>\n

Poursuivez votre apprentissage avec notre cours sp\u00e9cialis\u00e9 sur les suites g\u00e9om\u00e9triques,<\/strong> con\u00e7u pour rendre leur ma\u00eetrise accessible et agr\u00e9able !<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n\n\n\n

Exemple des suites g\u00e9om\u00e9triques<\/h2>\n\n\n\n

D\u00e9finition : suite g\u00e9om\u00e9trique<\/h3>\n\n\n\n

<\/p>\nOn appelle suite g\u00e9om\u00e9trique toute suite de la forme \"\left( a q^n \right)_{n \in \mathbb{N}}\" avec \"q \in \mathbb{R}\" et \"a \in \mathbb{R}\".\n\n\n\n

Proposition<\/h3>\n\n\n\n

<\/p>\nSoit \"q \in \mathbb{R}\". La suite \"\left( q^n \right)_{n \in \mathbb{N}} \begin{cases} \text{diverge vers}\; + \infty \; \text{si} \; q > 1 \\ \text{converge vers}\; 1 \; \text{si} \; q = 1 \\ \text{converge vers}\; 0 \; \text{si} \; \left| q \right| < 1 \\ \text{diverge sinon} \end{cases}\".\n\n\n\n

D\u00e9monstration<\/h3>\n\n\n\n

<\/p>\n

  • Une simple \u00e9tude de fonction permet de montrer que \n

      <\/span>   <\/span>\"\[\forall x > -1, \; \forall n \in \mathbb{N}, \quad \left( 1 + x \right)^n \ge 1 + n x.\]\"<\/p>\nSoit \"q>1\". Pour tout \"n \in \mathbb{N}\", on a \"q^n = \left( 1 + q - 1 \right)^n \ge 1 + n \left( q - 1 \right) \xrightarrow[n \to + \infty]{}+ \infty\". Par comparaison, on en d\u00e9duit que \"\left( q^n \right)_{n \in \mathbb{N}}\" diverge vers \"+ \infty\". <\/li>\n

  • Si \"q=1\", la suite \"\left( q^n \right)_{n \in \mathbb{N}}\" est constante \u00e9gale \u00e0 \"1\", donc converge et de limite \u00e9gale \u00e0 \"1\". <\/li>\n
  • On suppose que \"\left| q \right| < 1\". Le cas o\u00f9 \"q=0\" est clair car la suite \"\left( q^n \right)_{n \in \mathbb{N}}\" stationne sur \"0\".
    \nOn suppose donc \"q \neq 0\". On a \n

      <\/span>   <\/span>\"\[\forall n \in \mathbb{N}, \quad \left| q^n \right| = \left| q\right |^n = \mathrm{e}^{n \ln \left( \left| q \right| \right)} \xrightarrow[n \to + \infty]{} 0\]\"<\/p>\ncar \"\ln \left( \left| q \right| \right) < 0\" et car \"\lim\limits_{u \to - \infty} \mathrm{e}^{u} = 0\". <\/li>\n

  • Si \"\left( q^n \right)_{n \in \mathbb{N}}\" converge vers une limite \"\ell\", alors \"\left( q^{n+1} \right)_{n \in \mathbb{N}}\" converge vers \"q \ell\". Par unicit\u00e9 de la limite, \"\ell\" v\u00e9rifie\n\"\ell = q \ell\", soit \"\ell = 0\", ce qui est exclu car pour tout \"n \in \mathbb{N}\", \"\left| q^n \right| \ge 1\".
    \nUn raisonnement analogue permet de montrer que la suite \"\left( q^n \right)_{n \in \mathbb{N}}\" ne peut pas diverger vers \"+ \infty\" ou \"- \infty\". <\/li>\n\n\n\n

    Exemple<\/h3>\n\n\n\n

    <\/p>\nLa suite \"\displaystyle{\left(\sum_{k=0}^n \left( \dfrac12 \right)^k \right)_{n \in \mathbb{N}}}\" converge vers \"2\" car \n

      <\/span>   <\/span>\"\[\forall n \in \mathbb{N}, \quad \sum_{k=0}^n \left( \dfrac12 \right)^k = \dfrac{1- 1/2^{n+1}}{1- 1/2} \underset{n \to + \infty}{\longrightarrow} 2.\]\"<\/p>\n\n\n\n

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    \"livre<\/figure>\n<\/div>\n\n\n\n
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    Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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    \n Tu as aim\u00e9 cet article ?<\/span>\n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

    Vous trouvez les suites g\u00e9om\u00e9triques complexes ? Un cours particulier de maths avec nos profs peut vous aider (…)<\/p>\n","protected":false},"author":158,"featured_media":244595,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-234971","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234971","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=234971"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234971\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244595"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=234971"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=234971"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=234971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}