{"id":11893,"date":"2014-05-23T15:34:59","date_gmt":"2014-05-23T14:34:59","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11893"},"modified":"2022-01-13T11:04:58","modified_gmt":"2022-01-13T11:04:58","slug":"prove-que-a-sucessao-e-um-infinitamente-grande-negativo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11893","title":{"rendered":"Prove que a sucess\u00e3o \u00e9 um infinitamente grande negativo"},"content":{"rendered":"<p><ul id='GTTabs_ul_11893' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11893' class='GTTabs_curr'><a  id=\"11893_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11893' ><a  id=\"11893_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_11893'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $\\left( {{b_n}} \\right)$ uma sucess\u00e3o tal que ${b_n} = \\frac{{3 &#8211; 4n}}{2}$.<\/p>\n<ol>\n<li>Prove que a sucess\u00e3o \u00e9 um infinitamente grande negativo, usando a defini\u00e7\u00e3o e sem usar a defini\u00e7\u00e3o.<\/li>\n<li>Determine a menor ordem a partir da qual os termos da sucess\u00e3o s\u00e3o inferiores a $ &#8211; 500$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11893' onClick='GTTabs_show(1,11893)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11893'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Seja $\\left( {{b_n}} \\right)$ uma sucess\u00e3o tal que ${b_n} = \\frac{{3 &#8211; 4n}}{2}$.<\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote>\n<p>Prove que a sucess\u00e3o \u00e9 um infinitamente grande negativo, usando a defini\u00e7\u00e3o e sem usar a defini\u00e7\u00e3o.<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>Determine a menor ordem a partir da qual os termos da sucess\u00e3o s\u00e3o inferiores a $ &#8211; 500$.<\/p>\n<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00ad<\/p>\n<ol>\n<li>A sucess\u00e3o $\\left( {{b_n}} \\right)$ \u00e9 um infinitamente grande negativo se e s\u00f3 se $\\left( { &#8211; {b_n}} \\right)$ for um infinitamente grande positivo.\n<p>Seja $M \\in {\\mathbb{R}^ + }$.<br \/>\nOra,<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; {b_n} &gt; M}&amp; \\Leftrightarrow &amp;{ &#8211; \\frac{{3 &#8211; 4n}}{2} &gt; M} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{{4n &#8211; 3}}{2} &gt; M} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n &gt; \\frac{{2M + 3}}{4}}<br \/>\n\\end{array}\\]<br \/>\nConclui-se que $\\forall M \\in {\\mathbb{R}^ + },\\,\\,\\exists p \\in \\mathbb{N}:\\,\\,n &gt; p \\Rightarrow\u00a0 &#8211; {b_n} &gt; M$.<\/p>\n<p>Logo, $\\left( { &#8211; {b_n}} \\right)$ \u00e9 um infinitamente grande positivo e, consequentemente, $\\left( {{b_n}} \\right)$ \u00e9 um infinitamente grande negativo.<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Sem usar a defini\u00e7\u00e3o<\/strong><\/span>:<\/p>\n<p>Consideremos a fun\u00e7\u00e3o afim:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{f:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{3 &#8211; 4x}}{2}}<br \/>\n\\end{array}\\]<br \/>\nOra, sabe-se que \\(\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f\\left( x \\right) =\u00a0 &#8211; \\infty \\).<br \/>\nComo a sucess\u00e3o $\\left( {{b_n}} \\right)$ \u00e9 a restri\u00e7\u00e3o de $f$ ao conjunto $\\mathbb{N}$, ent\u00e3o ser\u00e1 $\\lim {b_n} =\u00a0 &#8211; \\infty $.<br \/>\nLogo, a sucess\u00e3o $\\left( {{b_n}} \\right)$ \u00e9 um infinitamente grande negativo.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Temos necessidade de resolver a condi\u00e7\u00e3o ${b_n} &lt;\u00a0 &#8211; 500 \\Leftrightarrow\u00a0 &#8211; {b_n} &gt; 500$, pelo que bastar\u00e1 considerar $M = 500$ na inequa\u00e7\u00e3o resolvida na al\u00ednea anterior:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{{b_n} &lt;\u00a0 &#8211; 500}&amp; \\Leftrightarrow &amp;{n &gt; \\frac{{2 \\times 500 + 3}}{4}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n &gt; 250,75}<br \/>\n\\end{array}\\]<br \/>\nPortanto, os termos da sucess\u00e3o $\\left( {{b_n}} \\right)$ s\u00e3o inferiores a $ &#8211; 500$ a partir da ordem $251$, inclusive.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11893' onClick='GTTabs_show(0,11893)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $\\left( {{b_n}} \\right)$ uma sucess\u00e3o tal que ${b_n} = \\frac{{3 &#8211; 4n}}{2}$. Prove que a sucess\u00e3o \u00e9 um infinitamente grande negativo, usando a defini\u00e7\u00e3o e sem usar a defini\u00e7\u00e3o. Determine&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14114,"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-11893","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":13342,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat56.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11893","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=11893"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11893\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14114"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11893"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}