{"id":11892,"date":"2014-05-23T15:02:03","date_gmt":"2014-05-23T14:02:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11892"},"modified":"2022-01-13T10:59:00","modified_gmt":"2022-01-13T10:59:00","slug":"prove-que-a-sucessao-e-um-infinitamente-grande-positivo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11892","title":{"rendered":"Prove que a sucess\u00e3o \u00e9 um infinitamente grande positivo"},"content":{"rendered":"<p><ul id='GTTabs_ul_11892' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11892' class='GTTabs_curr'><a  id=\"11892_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11892' ><a  id=\"11892_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_11892'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a sucess\u00e3o de termo geral ${a_n} = {n^2} + 1$.<\/p>\n<p>Prove que a sucess\u00e3o \u00e9 um infinitamente grande positivo:<\/p>\n<ol>\n<li>usando a defini\u00e7\u00e3o;<\/li>\n<li>sem usar a defini\u00e7\u00e3o.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11892' onClick='GTTabs_show(1,11892)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11892'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Considere a sucess\u00e3o de termo geral ${a_n} = {n^2} + 1$.<\/p>\n<p>Prove que a sucess\u00e3o \u00e9 um infinitamente grande positivo:<\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote>\n<p>usando a defini\u00e7\u00e3o;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>sem usar a defini\u00e7\u00e3o.<\/p>\n<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00ad<\/p>\n<ol>\n<li>Seja $M \\in {\\mathbb{R}^ + }$.<br \/>\nOra,<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{{a_n} &gt; M}&amp; \\Leftrightarrow &amp;{{n^2} + 1 &gt; M} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{n^2} &gt; M &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left( {\\begin{array}{*{20}{l}}<br \/>\n{n \\in \\mathbb{N}}&amp; \\wedge &amp;{M \\in \\left] {0,1} \\right[}<br \/>\n\\end{array}} \\right)}&amp; \\vee &amp;{\\left( {\\begin{array}{*{20}{l}}<br \/>\n{n &gt; \\sqrt {M &#8211; 1} }&amp; \\wedge &amp;{M \\in \\left[ {1, + \\infty } \\right[}<br \/>\n\\end{array}} \\right)}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<br \/>\nPelo exposto, conclui-se que $\\forall M \\in {\\mathbb{R}^ + },\\,\\,\\exists p \\in \\mathbb{N}:\\,\\,n &gt; p \\Rightarrow {a_n} &gt; M$.<\/p>\n<p>Logo, a sucess\u00e3o $\\left( {{a_n}} \\right)$ \u00e9 um infinitamente grande positivo.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Consideremos a seguinte fun\u00e7\u00e3o quadr\u00e1tica: \\[\\begin{array}{*{20}{l}}<br \/>\n{f:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to {x^2} + 1}<br \/>\n\\end{array}\\]<br \/>\nOra, sabe-se que \\[\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f\\left( x \\right) =\u00a0 + \\infty \\]<br \/>\nComo a sucess\u00e3o $\\left( {{a_n}} \\right)$ \u00e9 a restri\u00e7\u00e3o da fun\u00e7\u00e3o $f$ ao conjunto $\\mathbb{N}$, ent\u00e3o ser\u00e1 $\\lim {a_n} =\u00a0 + \\infty $.<br \/>\nLogo, a sucess\u00e3o $\\left( {{a_n}} \\right)$ \u00e9 um infinitamente grande positivo.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11892' onClick='GTTabs_show(0,11892)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere a sucess\u00e3o de termo geral ${a_n} = {n^2} + 1$. Prove que a sucess\u00e3o \u00e9 um infinitamente grande positivo: usando a defini\u00e7\u00e3o; sem usar a defini\u00e7\u00e3o. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19189,"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-11892","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":14433,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat75.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11892","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=11892"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11892\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19189"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11892"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11892"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11892"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11892"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}