{"id":2343,"date":"2020-05-03T16:36:49","date_gmt":"2020-05-03T14:36:49","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?page_id=2343"},"modified":"2023-04-16T16:20:21","modified_gmt":"2023-04-16T14:20:21","slug":"python-algorithme-de-seuil","status":"publish","type":"page","link":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/","title":{"rendered":"Python : algorithme de seuil"},"content":{"rendered":"\n

Un algorithme de seuil<\/em> est un algorithme ayant pour objectif de trouver le premier entier \\(n_0\\) \u00e0 partir duquel tous les termes de la suite \\((u_n)\\) satisfont une condition impos\u00e9e.<\/p>\n\n\n\n

\"Algorithme
Algorithme de seuil en Python<\/figcaption><\/figure><\/div>\n\n\n\n

Exemple 1 : suite d\u00e9finie de fa\u00e7on explicite<\/h2>\n\n\n\n

On consid\u00e8re la suite \\((u_n)\\) d\u00e9finie pour tout entier naturel n<\/em> par l’\u00e9galit\u00e9:$$u_n=\\frac{n^2+3x+5}{n^2+1}.$$On cherche le premier entier \\(n_0\\) \u00e0 partir duquel \\(u_n \\leqslant 1,1\\).<\/p>\n\n\n\n

Le programme suivant peut alors \u00eatre consid\u00e9r\u00e9:<\/p>\n\n\n\n

# on d\u00e9finit une fonction u(n) qui retourne la valeur de u(n)\ndef u(n):\n    return (n*n + 3*n + 5) \/ (n*n + 1)\n\n# ici commence le programme de seuil\nn = 0\nwhile u(n) > 1.1:\n    n = n + 1\n    \nprint(\"Le premier entier 'n' tel que u(n) <= 1.1 est : {}\".format(n))<\/pre>\n\n\n\n
Ce programme affiche la valeur 32 car \\(u_{31}\\approx1.1008>1.1\\) (donc la condition de la boucle while<\/em> est vraie, ce qui a pour cons\u00e9quence d’entrer dans la boucle et d’ajouter 1 \u00e0 n<\/em>), et comme \\(u_{32}\\approx 1.09756<1.1\\), la condition de la boucle while<\/em> n’est plus satisfaite, ce qui fait que l’on sort de la boucle. La valeur stock\u00e9e dans n<\/em> est donc 32.<\/p>\n\n\n\n
Exemple 2 : suite d\u00e9finie par r\u00e9currence<\/h2>\n\n\n\n
On consid\u00e8re la suite \\((v_n)\\) d\u00e9finie par son premier terme \\(v_0=7\\) et par la relation de r\u00e9currence:$$v_{n+1}=\\frac{1}{2}\\left(v_n+\\frac{5}{v_n}\\right)$$pour tout entier naturel n<\/em>.<\/p>\n\n\n\n
D\u00e9terminons le premier indice \u00e0 partir duquel \\(v_n \\leqslant 2.24\\). On utilise alors le programme suivant:<\/p>\n\n\n\n
 # on commence par initialiser les variables\nn = 0\nv = 7\n\n# ici commence le programme de seuil\nwhile v > 2.24:\n    n = n + 1 # l'indice du terme est incr\u00e9ment\u00e9 de 1\n    v = 0.5 * (v + 5 \/ v) # on utilise la relation de r\u00e9currence pour calculer le terme suivant\n    print(v)\n    \nprint(\"Le premier entier 'n' tel que v(n) <= 2.24 est : {}\".format(n))\n<\/pre>\n\n\n\n
Il affiche la valeur 4. En effet, \\(v_3\\approx 2.26 > 2.24\\) alors que \\(v_4\\approx 2.236\\).<\/p>\n\n\n\n
N’oubliez pas que si vous avez besoin d’un cours de maths, vous pouvez r\u00e9server un cours sur cette page<\/a>.<\/p>\n\n\n\n
[Retour \u00e0 la page pr\u00e9c\u00e9dente]<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
Un algorithme de seuil est un algorithme ayant pour objectif de trouver le premier entier \\(n_0\\) \u00e0 partir duquel tous les termes de la suite \\((u_n)\\) satisfont une condition impos\u00e9e. Exemple 1 : suite d\u00e9finie de fa\u00e7on explicite On consid\u00e8re la suite \\((u_n)\\) d\u00e9finie pour tout entier naturel n par […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2343","page","type-page","status-publish","hentry"],"yoast_head":"\nPython : algorithme de seuil - Mathweb.fr - Avec explications<\/title>\n<meta name=\"description\" content=\"Vous trouverez sur cette page des exemples de programmes Python avec algorithme de seuil sur les suites num\u00e9riques, avec explications.\" \/>\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.mathweb.fr\/euclide\/python-algorithme-de-seuil\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Python : algorithme de seuil - Mathweb.fr - Avec explications\" \/>\n<meta property=\"og:description\" content=\"Vous trouverez sur cette page des exemples de programmes Python avec algorithme de seuil sur les suites num\u00e9riques, avec explications.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:modified_time\" content=\"2023-04-16T14:20:21+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/06\/Python-suite-algorithme-seuil-1024x640.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data1\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/\",\"name\":\"Python : algorithme de seuil - Mathweb.fr - Avec explications\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/06\\\/Python-suite-algorithme-seuil-1024x640.png\",\"datePublished\":\"2020-05-03T14:36:49+00:00\",\"dateModified\":\"2023-04-16T14:20:21+00:00\",\"description\":\"Vous trouverez sur cette page des exemples de programmes Python avec algorithme de seuil sur les suites num\u00e9riques, avec explications.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/#primaryimage\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/06\\\/Python-suite-algorithme-seuil.png\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/06\\\/Python-suite-algorithme-seuil.png\",\"width\":1080,\"height\":675},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/python-algorithme-de-seuil\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Python : algorithme de seuil\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\",\"name\":\"Mathweb.fr\",\"description\":\"Math\u00e9matiques, LaTeX et Python\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"fr-FR\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\",\"name\":\"St\u00e9phane Pasquet\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"width\":74,\"height\":77,\"caption\":\"St\u00e9phane Pasquet\"},\"logo\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Python : algorithme de seuil - Mathweb.fr - Avec explications","description":"Vous trouverez sur cette page des exemples de programmes Python avec algorithme de seuil sur les suites num\u00e9riques, avec explications.","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.mathweb.fr\/euclide\/python-algorithme-de-seuil\/","og_locale":"fr_FR","og_type":"article","og_title":"Python : algorithme de seuil - Mathweb.fr - Avec explications","og_description":"Vous trouverez sur cette page des exemples de programmes Python avec algorithme de seuil sur les suites num\u00e9riques, avec explications.","og_url":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/","og_site_name":"Mathweb.fr","article_modified_time":"2023-04-16T14:20:21+00:00","og_image":[{"url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/06\/Python-suite-algorithme-seuil-1024x640.png","type":"","width":"","height":""}],"twitter_card":"summary_large_image","twitter_misc":{"Dur\u00e9e de lecture estim\u00e9e":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/","url":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/","name":"Python : algorithme de seuil - Mathweb.fr - Avec explications","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/#primaryimage"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/06\/Python-suite-algorithme-seuil-1024x640.png","datePublished":"2020-05-03T14:36:49+00:00","dateModified":"2023-04-16T14:20:21+00:00","description":"Vous trouverez sur cette page des exemples de programmes Python avec algorithme de seuil sur les suites num\u00e9riques, avec explications.","breadcrumb":{"@id":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/"]}]},{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/#primaryimage","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/06\/Python-suite-algorithme-seuil.png","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/06\/Python-suite-algorithme-seuil.png","width":1080,"height":675},{"@type":"BreadcrumbList","@id":"https:\/\/www.mathweb.fr\/euclide\/python-algorithme-de-seuil\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.mathweb.fr\/euclide\/"},{"@type":"ListItem","position":2,"name":"Python : algorithme de seuil"}]},{"@type":"WebSite","@id":"https:\/\/www.mathweb.fr\/euclide\/#website","url":"https:\/\/www.mathweb.fr\/euclide\/","name":"Mathweb.fr","description":"Math\u00e9matiques, LaTeX et Python","publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.mathweb.fr\/euclide\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":["Person","Organization"],"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69","name":"St\u00e9phane Pasquet","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","width":74,"height":77,"caption":"St\u00e9phane Pasquet"},"logo":{"@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp"}}]}},"_links":{"self":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/pages\/2343","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/comments?post=2343"}],"version-history":[{"count":0,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/pages\/2343\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media?parent=2343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}