{"id":11897,"date":"2014-05-24T00:36:14","date_gmt":"2014-05-23T23:36:14","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11897"},"modified":"2022-01-13T11:18:36","modified_gmt":"2022-01-13T11:18:36","slug":"prove-qua-a-sucessao-de-termo-geral-u_n-1-left-1-rightn-nao-e-um-infinitamente-pequeno","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11897","title":{"rendered":"Prove que a sucess\u00e3o de termo geral ${u_n} = 1 &#8211; {\\left( { &#8211; 1} \\right)^n}$ n\u00e3o \u00e9 um infinitamente pequeno"},"content":{"rendered":"<p><ul id='GTTabs_ul_11897' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11897' class='GTTabs_curr'><a  id=\"11897_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11897' ><a  id=\"11897_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11897'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Prove qua a sucess\u00e3o de termo geral ${u_n} = 1 &#8211; {\\left( { &#8211; 1} \\right)^n}$ n\u00e3o \u00e9 um infinitamente pequeno.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11897' onClick='GTTabs_show(1,11897)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11897'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Prove qua a sucess\u00e3o de termo geral ${u_n} = 1 &#8211; {\\left( { &#8211; 1} \\right)^n}$ n\u00e3o \u00e9 um infinitamente pequeno.<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>A sucess\u00e3o $\\left( {{u_n}} \\right)$ \u00e9 um infinitamente pequeno se e s\u00f3 se $\\forall \\delta\u00a0 \\in {\\mathbb{R}^ + },\\,\\,\\exists p \\in \\mathbb{N}:\\,\\,n &gt; p \\Rightarrow \\left| {{u_n}} \\right| &lt; \\delta $.<\/p>\n<p>Ora,<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{\\left| {{u_n}} \\right| &lt; \\delta }&amp; \\Leftrightarrow &amp;{\\left| {1 &#8211; {{\\left( { &#8211; 1} \\right)}^n}} \\right| &lt; \\delta } \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{\\left| {1 + 1} \\right| &lt; \\delta } \\\\<br \/>\n{n{\\text{ \u00edmpar}}}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{\\left| {1 &#8211; 1} \\right| &lt; \\delta } \\\\<br \/>\n{n{\\text{ par}}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{\\delta\u00a0 &gt; 2} \\\\<br \/>\n{n{\\text{ \u00edmpar}}}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{\\delta\u00a0 &gt; 0} \\\\<br \/>\n{n{\\text{ par}}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<\/p>\n<p>Verifica-se que apenas para $\\delta\u00a0 \\in \\left] {2, + \\infty } \\right[$ existe uma ordem\u00a0$p \\in \\mathbb{N}$ tal que $n &gt; p \\Rightarrow \\left| {{u_n}} \\right| &lt; \\delta $.<\/p>\n<p>Consequentemente, a sucess\u00e3o $\\left( {{u_n}} \\right)$ n\u00e3o \u00e9 um infinitamente pequeno.<\/p>\n<p>\u00ad<\/p>\n<p>Note que a conclus\u00e3o \u00e9 \u00f3bvia, j\u00e1 que: \\[{u_n} = 1 &#8211; {\\left( { &#8211; 1} \\right)^n} = \\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n2&amp; \\Leftarrow &amp;{n{\\text{ \u00edmpar}}} \\\\<br \/>\n0&amp; \\Leftarrow &amp;{n{\\text{ par}}}<br \/>\n\\end{array}} \\right.\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11897' onClick='GTTabs_show(0,11897)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Prove qua a sucess\u00e3o de termo geral ${u_n} = 1 &#8211; {\\left( { &#8211; 1} \\right)^n}$ n\u00e3o \u00e9 um infinitamente pequeno. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":14060,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,372],"tags":[422,376,431],"series":[],"class_list":["post-11897","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-sucessoes-reais","tag-11-o-ano","tag-infinitesimo","tag-sucessoes-reais"],"views":3165,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat05.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11897","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11897"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11897\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14060"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11897"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11897"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11897"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11897"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}