{"id":1851,"date":"2017-06-01T13:29:12","date_gmt":"2017-06-01T11:29:12","guid":{"rendered":"http:\/\/www.mathnique.com\/site\/?page_id=1851"},"modified":"2018-07-25T13:04:03","modified_gmt":"2018-07-25T11:04:03","slug":"generalites-sur-les-series","status":"publish","type":"page","link":"https:\/\/www.mathnique.com\/site\/generalites-sur-les-series\/","title":{"rendered":"G\u00e9n\u00e9ralit\u00e9s sur les s\u00e9ries"},"content":{"rendered":"<ul>\n<li><span style=\"color: #ff0000;\"><strong>Cours personnel sur les s\u00e9ries<\/strong><\/span> : disponible bient\u00f4t<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2522 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2018\/07\/tartelehning-300x235.png\" alt=\"\" width=\"300\" height=\"235\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2018\/07\/tartelehning-300x235.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2018\/07\/tartelehning-768x601.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2018\/07\/tartelehning-1024x801.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2018\/07\/tartelehning-1200x939.png 1200w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2018\/07\/tartelehning.png 1230w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<li><span style=\"color: #ff0000;\"><strong>Probl\u00e8mes classiques<\/strong> :<\/span>\n<ul>\n<li><span style=\"color: #ff0000;\"><strong><em>la s\u00e9rie harmonique<\/em><\/strong><\/span>\u00a0<strong><em><span style=\"color: #ff0000;\">divergente<\/span><\/em><\/strong> $\\zeta_1$ = $\\displaystyle{\\sum_{k = 0}^{+\\infty} \\dfrac{1}{n}}$<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1842 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/serieharmonique-135x300.jpg\" alt=\"\" width=\"144\" height=\"320\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/serieharmonique-135x300.jpg 135w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/serieharmonique.jpg 146w\" sizes=\"auto, (max-width: 144px) 85vw, 144px\" \/><br \/>\nL'\u00e9v\u00eaque Nicole d'Oresme(1325-1382) ma\u00eetre d'oeuvre de construction d'\u00e9glises a\u00a0\u00e9tudi\u00e9 les empilements de briques sans mortier r\u00e9alis\u00e9s \u00a0par les ma\u00e7ons de l'\u00e9poque. il a compris que pour garder la structure d'une vo\u00fbte\u00a0en \u00e9quilibre , il faut que le centre de gravit\u00e9 de la brique sup\u00e9rieure soit encore au dessus de la brique inf\u00e9rieure :<br \/>\nSi l'on veut placer par exemple 2 briques sur une 3\u00e8me, le centre de gravit\u00e9 commun $G_2$ des deux pr\u00e9c\u00e9dentes se trouve au bord de la 3\u00e8me donc du bord droit de la brique 1 \u00e0<br \/>\n$\\dfrac{(1 \\times \\dfrac{1}{2} + 1 \\times 1)}{2} = \\dfrac{3}{4}$ d'unit\u00e9.<br \/>\nLa brique 4 est \u00e0 placer au centre de gravit\u00e9 $G_3$ des 3 sup\u00e9rieures, c'est-\u00e0-dire avec un d\u00e9calage de<br \/>\n$\\dfrac{(2 \\times \\dfrac{3}{4} + 1 \\times \\dfrac{5}{4})}{2} = \\dfrac{11}{12}$ brique.<br \/>\nLes d\u00e9calages successifs sont $\\dfrac{1}{2} +\u00a0\\dfrac{1}{4} +\\dfrac{1}{6} +\\dfrac{1}{8} + \\cdots =<br \/>\n2( \\dfrac{1}{1} +\u00a0\\dfrac{1}{2} +\u00a0\\dfrac{1}{3} +\\dfrac{1}{4} +\\dfrac{1}{5} + \\cdots\u00a0) $<br \/>\nNicole d'Oresme a d\u00e9montr\u00e9 que la s\u00e9rie harmonique\u00a0est divergente<br \/>\n<em>(Source : extrait du livre La Physique de tous les jours- Istvan Berkes - Editions Vuibert)<br \/>\n<\/em>Ce probl\u00e8me d'empilement est tr\u00e8s bien expliqu\u00e9 sur le site suivant :<br \/>\n<a href=\"http:\/\/prof.pantaloni.free.fr\/IMG\/pdf\/Surplomb-de-briques.pdf\">http:\/\/prof.pantaloni.free.fr\/IMG\/pdf\/Surplomb-de-briques.pdf<\/a><\/li>\n<li><span style=\"color: #ff0000;\">Cours personnel plus exercices<\/span> :\u00a0<a href=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/serieharmonique.pdf\">serieharmonique<\/a>\n<ul>\n<li><em><strong><span style=\"color: #ff0000;\">la s\u00e9rie<\/span>\u00a0<span style=\"color: #ff0000;\">convergente<\/span> $\\zeta_2$ =<\/strong><\/em>\u00a0$\\displaystyle{\\sum_{k = 0}^{+\\infty} \\dfrac{1}{n^2} = \\dfrac{\\pi^2}{6}}$ :\u00a0<a href=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/PbDzeta2.pdf\">PbDzeta2<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><span style=\"color: #ff0000;\"><em><strong>Vid\u00e9o des Amphis de la 5<\/strong><\/em><\/span> (Professeur Jacques VAUTHIER) sur les s\u00e9ries :<br \/>\n<a href=\"https:\/\/youtu.be\/fuh5sEIfLn8?t=96\">https:\/\/youtu.be\/fuh5sEIfLn8?t=96<\/a><\/li>\n<li><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Cours personnel sur les s\u00e9ries : disponible bient\u00f4t Probl\u00e8mes classiques : la s\u00e9rie harmonique\u00a0divergente $\\zeta_1$ = $\\displaystyle{\\sum_{k = 0}^{+\\infty} \\dfrac{1}{n}}$ L'\u00e9v\u00eaque Nicole d'Oresme(1325-1382) ma\u00eetre d'oeuvre de construction d'\u00e9glises a\u00a0\u00e9tudi\u00e9 les empilements de briques sans mortier r\u00e9alis\u00e9s \u00a0par les ma\u00e7ons de l'\u00e9poque. il a compris que pour garder la structure d'une vo\u00fbte\u00a0en \u00e9quilibre , il faut &hellip; <a href=\"https:\/\/www.mathnique.com\/site\/generalites-sur-les-series\/\" class=\"more-link\">Continuer la lecture<span class=\"screen-reader-text\"> de &laquo;&nbsp;G\u00e9n\u00e9ralit\u00e9s sur les s\u00e9ries&nbsp;&raquo;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-1851","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/1851","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/comments?post=1851"}],"version-history":[{"count":11,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/1851\/revisions"}],"predecessor-version":[{"id":2532,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/1851\/revisions\/2532"}],"wp:attachment":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/media?parent=1851"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}