{"id":17867,"date":"2016-09-17T16:04:11","date_gmt":"2016-09-17T16:04:11","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=17867"},"modified":"2016-09-17T16:04:11","modified_gmt":"2016-09-17T16:04:11","slug":"calculus-ratio-test-checking-%cf%834nn-for-convergence","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/calculus-ratio-test-checking-%cf%834nn-for-convergence\/","title":{"rendered":"Calculus:  ratio test:  checking \u03a34<sup>n<\/sup>\/n! for convergence"},"content":{"rendered":"<h1>The tutor uses the ratio test to show the infinite series \u03a34<sup>n<\/sup>\/n! converges.<\/h1>\n<p><strong>Example:  Check \u03a3<sub>0<\/sub><sup>\u221e<\/sup>4<sup>n<\/sup>\/n! for convergence or divergence.<\/h1>\n<p><\/strong><\/p>\n<p>Solution:  The ratio test says that, if lim<sub>n\u2192\u221e<\/sub>|a<sub>n+1<\/sub>\/a<sub>n<\/sub>| &#60; 1, then the series converges. In this case, the terms are all positive anyway, so lim<sub>n\u2192\u221e<\/sub>a<sub>n+1<\/sub>\/a<sub>n<\/sub> &#60; 1 will indicate convergence:<\/p>\n<p>a<sub>n+1<\/sub>\/a<sub>n<\/sub> = (4<sup>n+1<\/sup>\/(n+1)!)\/(4<sup>n<\/sup>\/n!)<\/p>\n<p>which becomes 4<sup>n+1<\/sup>\/(n+1)! * n!\/4<sup>n<\/sup><\/p>\n<p>which simplifies to 4\/(n+1).<\/p>\n<p>lim<sub>n\u2192\u221e<\/sub>4\/(n+1) = 0 &#60; 1, so the series \u03a3<sub>0<\/sub><sup>\u221e<\/sup>4<sup>n<\/sup>\/n! converges.<\/p>\n<p>Source:<\/p>\n<p>Larson, Roland E. and Robert P. Hostetler.  <u>Calculus,<\/u> third ed.  Toronto:  D C Heath and Company, 1989.<\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The tutor uses the ratio test to show the infinite series \u03a34n\/n! converges. Example: Check \u03a30\u221e4n\/n! for convergence or divergence. Solution: The ratio test says that, if limn\u2192\u221e|an+1\/an| &#60; 1, then the series converges. In this case, the terms are &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/calculus-ratio-test-checking-%cf%834nn-for-convergence\/\"> <span class=\"screen-reader-text\">Calculus:  ratio test:  checking \u03a34<sup>n<\/sup>\/n! for convergence<\/span> Read More &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":[234],"tags":[1893,1892],"class_list":["post-17867","post","type-post","status-publish","format-standard","hentry","category-calculus","tag-checking-infinite-series-for-convergence","tag-ratio-test"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17867","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=17867"}],"version-history":[{"count":11,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17867\/revisions"}],"predecessor-version":[{"id":17878,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17867\/revisions\/17878"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=17867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=17867"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=17867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}