{"id":11894,"date":"2014-05-23T19:26:28","date_gmt":"2014-05-23T18:26:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11894"},"modified":"2022-01-13T11:08:40","modified_gmt":"2022-01-13T11:08:40","slug":"averigue-se-a-sucessao-e-um-infinitamente-grande","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11894","title":{"rendered":"Averigue se a sucess\u00e3o \u00e9 um infinitamente grande"},"content":{"rendered":"<p><ul id='GTTabs_ul_11894' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11894' class='GTTabs_curr'><a  id=\"11894_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11894' ><a  id=\"11894_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_11894'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a seguinte afirma\u00e7\u00e3o:<\/p>\n<p>&#8220;A sucess\u00e3o de termo geral ${c_n} = {\\left( { &#8211; 1} \\right)^n}{n^3}$ \u00e9 um infinitamente grande.&#8221;<\/p>\n<p>Averigue se esta afirma\u00e7\u00e3o \u00e9 verdadeira ou falsa.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11894' onClick='GTTabs_show(1,11894)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11894'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Considere a seguinte afirma\u00e7\u00e3o:<\/p>\n<p>&#8220;A sucess\u00e3o de termo geral ${c_n} = {\\left( { &#8211; 1} \\right)^n}{n^3}$ \u00e9 um infinitamente grande.&#8221;<\/p>\n<p>Averigue se esta afirma\u00e7\u00e3o \u00e9 verdadeira ou falsa.<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>A sucess\u00e3o $\\left( {{c_n}} \\right)$ \u00e9 um infinitamente grande se e s\u00f3 se $\\left( {\\left| {{c_n}} \\right|} \\right)$ for um infinitamente grande positivo.<\/p>\n<p>Seja $M \\in {\\mathbb{R}^ + }$.<\/p>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{\\left| {{c_n}} \\right| &gt; M}&amp; \\Leftrightarrow &amp;{\\left| {{{\\left( { &#8211; 1} \\right)}^n}{n^3}} \\right| &gt; M} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{n^3} &gt; M} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n &gt; \\sqrt[3]{M}}<br \/>\n\\end{array}\\]<\/p>\n<p>Conclui-se que $\\forall M \\in {\\mathbb{R}^ + },\\,\\,\\exists p \\in \\mathbb{N}:\\,\\,n &gt; p \\Rightarrow \\left| {{c_n}} \\right| &gt; M$.<\/p>\n<p>Logo, a sucess\u00e3o $\\left( {\\left| {{c_n}} \\right|} \\right)$ \u00e9 um infinitamente grande positivo e, consequentemente, $\\left( {{c_n}} \\right)$ \u00e9 um infinitamente grande.<\/p>\n<p>Portanto, a afirma\u00e7\u00e3o \u00e9 verdadeira.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11894' onClick='GTTabs_show(0,11894)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere a seguinte afirma\u00e7\u00e3o: &#8220;A sucess\u00e3o de termo geral ${c_n} = {\\left( { &#8211; 1} \\right)^n}{n^3}$ \u00e9 um infinitamente grande.&#8221; Averigue se esta afirma\u00e7\u00e3o \u00e9 verdadeira ou falsa. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14057,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,372],"tags":[422,375,431],"series":[],"class_list":["post-11894","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-sucessoes-reais","tag-11-o-ano","tag-infinitamente-grande","tag-sucessoes-reais"],"views":2009,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat02.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11894","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=11894"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11894\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14057"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11894"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}