{"id":2519,"date":"2012-09-24T14:05:05","date_gmt":"2012-09-24T06:05:05","guid":{"rendered":"http:\/\/vanabel.sinaapp.com\/?p=2519"},"modified":"2012-09-24T14:05:05","modified_gmt":"2012-09-24T06:05:05","slug":"a-problem-about-limits-of-sequence","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2519","title":{"rendered":"A Problem about Limits of Sequence"},"content":{"rendered":"<p>I have been asked to solve the following problem:<br \/>\nLet $\\set{x_n}_{n=1}^\\infty$ be a real sequence defined by $$x_{n+1}=\\frac{C}{2}+\\frac{x_n^2}{2},$$ \u00a0with $x_1=C\/2$, where $C$ is a constant, try to show that<\/p>\n<ol>\n<li>If $C&gt;1$, then $\\set{x_n}_{n=1}^\\infty$ is divergent;<\/li>\n<li>If $0&lt; C\\leq 1$, then $\\set{x_n}_{n=1}^\\infty$ is convergent;<\/li>\n<li>If $-3\\leq C &lt; 0$, then $\\set{x_n}_{n=1}^\\infty$ is convergent;<\/li>\n<\/ol>\n<p>Try to discuss the case of $C&lt; -3$, is $\\set{x_n}_{n=1}^\\infty$ divergent?<\/p>\n<p>If you have any idea, please tell me! Just leave a word below!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I have been asked to solve the following &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=2519\"> <span class=\"screen-reader-text\">A Problem about Limits of Sequence<\/span> \u9605\u8bfb\u66f4\u591a &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[539,619,789,803,800],"class_list":["post-2519","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-limits","tag-sequence","tag-789","tag-803","tag-lecture-notes"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2519","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2519"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2519\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2519"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2519"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2519"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}