{"id":57,"date":"2024-02-03T12:00:00","date_gmt":"2024-02-03T11:00:00","guid":{"rendered":"https:\/\/www.maths-sup.com\/?p=57"},"modified":"2024-02-03T18:30:53","modified_gmt":"2024-02-03T17:30:53","slug":"introduction-aux-series-entieres","status":"publish","type":"post","link":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/","title":{"rendered":"Introduction aux S\u00e9ries enti\u00e8res"},"content":{"rendered":"\n

Les s\u00e9ries enti\u00e8res<\/strong> constituent une cat\u00e9gorie essentielle dans l’univers des s\u00e9ries de fonctions<\/strong>.
Elles jouent un r\u00f4le crucial dans de nombreux domaines des math\u00e9matiques.<\/p>\n\n\n\n

Cet article vise \u00e0 pr\u00e9senter les concepts fondamentaux des s\u00e9ries enti\u00e8res, en s’adressant particuli\u00e8rement \u00e0 ceux qui souhaitent apprendre avec un minimum de pr\u00e9requis.<\/p>\n\n\n\n

Dans cet article, nous pr\u00e9sentons toutes les notions importantes \u00e0 conna\u00eetre sur les s\u00e9ries enti\u00e8res.<\/strong><\/p>\n\n\n\n

Qu’est-ce qu’une s\u00e9rie enti\u00e8re?<\/strong><\/h2>\n\n\n\n

Pour comprendre les s\u00e9ries enti\u00e8res<\/strong>, il est essentiel de rappeler d’abord ce qu’est une s\u00e9rie de fonctions<\/strong>.<\/p>\n\n\n\n

Une s\u00e9rie de fonctions<\/strong> est la donn\u00e9e de deux suites de fonctions (S_n)_{n \\in \\mathbb{N}}<\/span> et (u_n)_{n \\in \\mathbb{N}}<\/span> telles que:<\/p>\n\n\n\n

\\displaystyle \\forall N \\in \\mathbb{N}, \\forall x, \\quad S_N(x) = \\sum_{n=0}^N u_n(x)<\/span><\/p>\n\n\n\n

En g\u00e9n\u00e9ral, on consid\u00e8re des s\u00e9ries de fonctions d\u00e9finies de \\mathbb{R}<\/span> vers \\mathbb{R}<\/span> ou bien \\mathbb{C}<\/span> vers \\mathbb{C}<\/span>, auquel cas on note parfois la variable z<\/span>.
On appelle (u_n)_{n \\in \\mathbb{N}}<\/span> le terme g\u00e9n\u00e9ral de la s\u00e9rie<\/strong>.
Une s\u00e9rie de terme g\u00e9n\u00e9ral (u_n)_{n \\in \\mathbb{N}}<\/span> est plus souvent not\u00e9e de mani\u00e8re synth\u00e9tique : \\sum u_n<\/span> ou \\sum u_n(x)<\/span>.<\/p>\n\n\n\n

D\u00e9finition d’une s\u00e9rie enti\u00e8re:<\/strong>
Une s\u00e9rie enti\u00e8re<\/strong> est une s\u00e9rie de fonctions de la forme:<\/p>\n\n\n\n

\\displaystyle \\sum_{n \\in \\mathbb{N}} a_n x^n,<\/span><\/p>\n\n\n\n

avec (a_n)_{n \\in \\mathbb{N}}<\/span> une suite de \\mathbb{R}<\/span> ou de \\mathbb{C}<\/span>.<\/p>\n\n\n\n

Une s\u00e9rie enti\u00e8re est donc en quelque sorte un \u00ab\u00a0polyn\u00f4me de degr\u00e9 infinie\u00a0\u00bb.<\/p>\n\n\n\n\n\n\n\n

Convergence d’une s\u00e9rie enti\u00e8re<\/h2>\n\n\n\n

Lorsqu’on a une s\u00e9rie ou une suite de fonctions, il est important de conna\u00eetre sa convergence<\/strong>.<\/p>\n\n\n\n

Le lemme suivant est premier r\u00e9sultat fondamental sur la convergence des s\u00e9ries enti\u00e8res<\/strong>.<\/p>\n\n\n\n

Lemme d’Abel:<\/strong>
Soit \\sum_{n \\in \\mathbb{N}} a_n x^n,<\/span> une s\u00e9rie enti\u00e8re \u00e0 valeurs dans \\mathbb{R}<\/span>.
Soit R>0<\/span>. On suppose que la suite (|a_n| \\cdot R^n)_{n}<\/span> est born\u00e9e.
Alors pour tout r<R<\/span>, la s\u00e9rie de fonctions \\sum_{n \\in \\mathbb{N}} a_n x^n,<\/span> converge normalement sur l’intervalle ferm\u00e9 [-r, r]<\/span>.<\/p>\n\n\n\n

Preuve du lemme d’Abel:<\/strong>
Soit r<R<\/span>.
Comme (|a_n| \\cdot R^n)_{n}<\/span> est born\u00e9e, il existe M>0<\/span> tel que:<\/p>\n\n\n\n

\\forall n \\geq 0, \\quad |a_n| \\cdot R^n < M.<\/span><\/p>\n\n\n\n

On a:<\/p>\n\n\n\n

\\forall n \\geq 0, \\quad \\mathrm{sup}_{x \\in [-r, r]} | a_n \\cdot x^n| =|a_n| r^n \\leq M \\cdot (r\/R)^n.<\/span><\/p>\n\n\n\n

On a r\/R < 1<\/span>, donc la s\u00e9rie g\u00e9om\u00e9trique \\sum_n M \\cdot (r\/R)^n<\/span> converge.
Par th\u00e9or\u00e8me de domination sur les s\u00e9ries<\/strong>, la s\u00e9rie \\sum_n \\mathrm{sup}_{x \\in [-r, r]} | a_n \\cdot x^n|<\/span> converge.<\/p>\n\n\n\n

Ce qui prouve que \\sum_n a_n \\cdot x^n<\/span> converge normalement sur [-r, r]<\/span>. \\square<\/span><\/p>\n\n\n\n

D\u00e9finition du rayon de convergence:<\/strong>
Soit \\sum_{n \\in \\mathbb{N}} a_n x^n,<\/span> une s\u00e9rie enti\u00e8re.
Son rayon de convergence<\/strong> est:<\/p>\n\n\n\n

\\mathrm{sup} \\{ r \\in \\mathbb{R}_{\\geq 0} \\quad | \\quad (|a_n| r^n)_{n \\geq 0} \\text{ est born\u00e9e} \\}<\/span><\/p>\n\n\n\n

On note de R<\/span> le rayon de convergence.
Alors le lemme d’Abel implique que:<\/p>\n\n\n\n

    \n
  • La s\u00e9rie converge simplement<\/strong> sur l’intervalle ouvert ]-R, R[<\/span>.<\/li>\n\n\n\n
  • La convergence est normale sur tout intervalle compact<\/strong>.<\/li>\n\n\n\n
  • La fonction limite est continue<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    D’autre part on peut voir que pour tout x<\/span> tel que |x|>R<\/span>, le terme g\u00e9n\u00e9ral de la s\u00e9rie num\u00e9rique \\sum_n a_n \\cdot x^n<\/span> est non-born\u00e9e<\/strong>.
    Donc en dehors du l’intervalle ferm\u00e9 [-R, R]<\/span> la s\u00e9rie enti\u00e8re diverge grossi\u00e8rement<\/strong>.<\/p>\n\n\n\n

    Exemples fondamentaux<\/h2>\n\n\n\n

    Exemple 1<\/h3>\n\n\n\n

    Prenons la s\u00e9rie enti\u00e8re la plus simple: \\sum_n x^n<\/span>.<\/p>\n\n\n\n

    D’apr\u00e8s la d\u00e9finition du rayon de convergence, on peut voir qu’il vaut R = 1<\/span>.
    Ainsi, on peut en d\u00e9duire que la s\u00e9rie converge vers une fonction continue sur l’intervalle ouvert ]-1,1[<\/span>.<\/p>\n\n\n\n

    La formule de somme des suites g\u00e9om\u00e9triques, montre que la fonction limite est x \\longmapsto \\frac{1}{1-x}<\/span>.<\/p>\n\n\n\n

    Exemple 2<\/h2>\n\n\n\n

    Prenons la s\u00e9rie enti\u00e8re: \\sum_n \\frac{x^n}{n!}<\/span>.<\/p>\n\n\n\n

    On pose, pour tout n \\geq 0<\/span>, a_n = \\frac{1}{n!}<\/span>.<\/p>\n\n\n\n

    Soit R>0<\/span>.
    Par th\u00e9or\u00e8me de comparaison, la suite (a_n \\cdot R^n)_{n \\geq 0}<\/span> tend vers 0, donc est born\u00e9e.<\/p>\n\n\n\n

    Ainsi le rayon de convergence de \\sum_n \\frac{x^n}{n!}<\/span> est R = + \\infty<\/span>.<\/p>\n\n\n\n

    Le s\u00e9rie converge donc vers une fonction continue sur \\mathbb{R}<\/span>.<\/p>\n\n\n\n

    Ce qu’il faut retenir<\/h2>\n\n\n\n
      \n
    • Une s\u00e9rie enti\u00e8re<\/strong> est une s\u00e9rie de fonctions de la forme: \\sum_n a_n \\cdot x^n<\/span>.<\/li>\n\n\n\n
    • Une s\u00e9rie enti\u00e8re admet un rayon de convergence<\/strong> R<\/span>.<\/li>\n\n\n\n
    • La s\u00e9rie converge<\/strong> sur ]-R, R[<\/span> vers une fonction continue<\/strong>.<\/li>\n<\/ul>\n\n\n\n

      N’h\u00e9site pas \u00e0 interagir et \u00e0 poser des questions en commentaire.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Les s\u00e9ries enti\u00e8res constituent une cat\u00e9gorie essentielle dans l’univers des s\u00e9ries de fonctions.Elles jouent un r\u00f4le crucial dans de nombreux domaines des math\u00e9matiques. Cet article vise \u00e0 pr\u00e9senter les concepts fondamentaux des s\u00e9ries enti\u00e8res, en s’adressant particuli\u00e8rement \u00e0 ceux qui souhaitent apprendre avec un minimum de pr\u00e9requis. Dans cet article, […]<\/p>\n","protected":false},"author":2,"featured_media":97,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[8,9,7],"tags":[],"class_list":["post-57","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-analyse","category-lemme-dabel","category-serie-entiere"],"yoast_head":"\nIntroduction aux S\u00e9ries enti\u00e8res - Maths-Sup<\/title>\n<meta name=\"description\" content=\"Plongez dans le monde des s\u00e9ries enti\u00e8res \ud83d\udcda\ud83d\udca1! Une intro simple et claire pour tous les passionn\u00e9s de maths \ud83e\uddee\ud83d\ude80. Cliquez ici!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Introduction aux S\u00e9ries enti\u00e8res - Maths-Sup\" \/>\n<meta property=\"og:description\" content=\"Plongez dans le monde des s\u00e9ries enti\u00e8res \ud83d\udcda\ud83d\udca1! Une intro simple et claire pour tous les passionn\u00e9s de maths \ud83e\uddee\ud83d\ude80. Cliquez ici!\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\" \/>\n<meta property=\"og:site_name\" content=\"Maths-Sup\" \/>\n<meta property=\"article:published_time\" content=\"2024-02-03T11:00:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-02-03T17:30:53+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor-1024x585.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"585\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"Auguste Hoang\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"Auguste Hoang\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\"},\"author\":{\"name\":\"Auguste Hoang\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25\"},\"headline\":\"Introduction aux S\u00e9ries enti\u00e8res\",\"datePublished\":\"2024-02-03T11:00:00+00:00\",\"dateModified\":\"2024-02-03T17:30:53+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\"},\"wordCount\":978,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/www.maths-sup.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png\",\"articleSection\":[\"analyse\",\"lemme d'abel\",\"s\u00e9rie enti\u00e8re\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\",\"url\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\",\"name\":\"Introduction aux S\u00e9ries enti\u00e8res - Maths-Sup\",\"isPartOf\":{\"@id\":\"https:\/\/www.maths-sup.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png\",\"datePublished\":\"2024-02-03T11:00:00+00:00\",\"dateModified\":\"2024-02-03T17:30:53+00:00\",\"description\":\"Plongez dans le monde des s\u00e9ries enti\u00e8res \ud83d\udcda\ud83d\udca1! Une intro simple et claire pour tous les passionn\u00e9s de maths \ud83e\uddee\ud83d\ude80. Cliquez ici!\",\"breadcrumb\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage\",\"url\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png\",\"contentUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png\",\"width\":1792,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\/\/www.maths-sup.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Introduction aux S\u00e9ries enti\u00e8res\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.maths-sup.com\/#website\",\"url\":\"https:\/\/www.maths-sup.com\/\",\"name\":\"Maths-Sup\",\"description\":\"Votre Guide Complet pour l'Agr\u00e9gation de Math\u00e9matiques\",\"publisher\":{\"@id\":\"https:\/\/www.maths-sup.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.maths-sup.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"fr-FR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.maths-sup.com\/#organization\",\"name\":\"Maths-Sup\",\"url\":\"https:\/\/www.maths-sup.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png\",\"contentUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png\",\"width\":1024,\"height\":1024,\"caption\":\"Maths-Sup\"},\"image\":{\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25\",\"name\":\"Auguste Hoang\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g\",\"caption\":\"Auguste Hoang\"},\"sameAs\":[\"https:\/\/www.maths-sup.com\"],\"url\":\"https:\/\/www.maths-sup.com\/index.php\/author\/auguste-hoang\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Introduction aux S\u00e9ries enti\u00e8res - Maths-Sup","description":"Plongez dans le monde des s\u00e9ries enti\u00e8res \ud83d\udcda\ud83d\udca1! Une intro simple et claire pour tous les passionn\u00e9s de maths \ud83e\uddee\ud83d\ude80. Cliquez ici!","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/","og_locale":"fr_FR","og_type":"article","og_title":"Introduction aux S\u00e9ries enti\u00e8res - Maths-Sup","og_description":"Plongez dans le monde des s\u00e9ries enti\u00e8res \ud83d\udcda\ud83d\udca1! Une intro simple et claire pour tous les passionn\u00e9s de maths \ud83e\uddee\ud83d\ude80. Cliquez ici!","og_url":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/","og_site_name":"Maths-Sup","article_published_time":"2024-02-03T11:00:00+00:00","article_modified_time":"2024-02-03T17:30:53+00:00","og_image":[{"width":1024,"height":585,"url":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor-1024x585.png","type":"image\/png"}],"author":"Auguste Hoang","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"Auguste Hoang","Dur\u00e9e de lecture estim\u00e9e":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#article","isPartOf":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/"},"author":{"name":"Auguste Hoang","@id":"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25"},"headline":"Introduction aux S\u00e9ries enti\u00e8res","datePublished":"2024-02-03T11:00:00+00:00","dateModified":"2024-02-03T17:30:53+00:00","mainEntityOfPage":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/"},"wordCount":978,"commentCount":0,"publisher":{"@id":"https:\/\/www.maths-sup.com\/#organization"},"image":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage"},"thumbnailUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png","articleSection":["analyse","lemme d'abel","s\u00e9rie enti\u00e8re"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/","url":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/","name":"Introduction aux S\u00e9ries enti\u00e8res - Maths-Sup","isPartOf":{"@id":"https:\/\/www.maths-sup.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage"},"image":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage"},"thumbnailUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png","datePublished":"2024-02-03T11:00:00+00:00","dateModified":"2024-02-03T17:30:53+00:00","description":"Plongez dans le monde des s\u00e9ries enti\u00e8res \ud83d\udcda\ud83d\udca1! Une intro simple et claire pour tous les passionn\u00e9s de maths \ud83e\uddee\ud83d\ude80. Cliquez ici!","breadcrumb":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/"]}]},{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#primaryimage","url":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png","contentUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/02\/DALL\u00b7E-2024-02-03-18.23.26-A-visually-striking-and-engaging-featured-image-for-a-math-blog-post-about-entire-series-designed-in-a-wide-format.-The-image-should-creatively-incor.png","width":1792,"height":1024},{"@type":"BreadcrumbList","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/02\/03\/introduction-aux-series-entieres\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.maths-sup.com\/"},{"@type":"ListItem","position":2,"name":"Introduction aux S\u00e9ries enti\u00e8res"}]},{"@type":"WebSite","@id":"https:\/\/www.maths-sup.com\/#website","url":"https:\/\/www.maths-sup.com\/","name":"Maths-Sup","description":"Votre Guide Complet pour l'Agr\u00e9gation de Math\u00e9matiques","publisher":{"@id":"https:\/\/www.maths-sup.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.maths-sup.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"fr-FR"},{"@type":"Organization","@id":"https:\/\/www.maths-sup.com\/#organization","name":"Maths-Sup","url":"https:\/\/www.maths-sup.com\/","logo":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/","url":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png","contentUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png","width":1024,"height":1024,"caption":"Maths-Sup"},"image":{"@id":"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25","name":"Auguste Hoang","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.maths-sup.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g","caption":"Auguste Hoang"},"sameAs":["https:\/\/www.maths-sup.com"],"url":"https:\/\/www.maths-sup.com\/index.php\/author\/auguste-hoang\/"}]}},"_links":{"self":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts\/57","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/comments?post=57"}],"version-history":[{"count":27,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts\/57\/revisions"}],"predecessor-version":[{"id":91,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts\/57\/revisions\/91"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/media\/97"}],"wp:attachment":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/media?parent=57"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/categories?post=57"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/tags?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}