{"id":12114,"date":"2015-09-02T21:36:33","date_gmt":"2015-09-02T21:36:33","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=12114"},"modified":"2018-02-13T19:08:12","modified_gmt":"2018-02-13T19:08:12","slug":"math-implications-from-the-telescoping-sum","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-implications-from-the-telescoping-sum\/","title":{"rendered":"Math:  implications from the telescoping sum"},"content":{"rendered":"<h1>The tutor follows the idea of the telescoping sum.<\/h1>\n<p>In my <a href=\"?p=12094\">previous post<\/a>, I gave an example of a telescoping sum.  It implies that<\/p>\n<p>1-x^(n+1)=(1-x)(1+x+x^2+x^3+x^4+&#8230;..+x^n)<\/p>\n<p>Dividing both sides by 1-x we arrive at<\/p>\n<p>(1-x^(n+1))\/(1-x)=1+x+x^2+x^3+&#8230;..+x^n<\/p>\n<p>which is familiar from the sum of a geometric series of n+1 terms.<\/p>\n<p>If -1&#60;x&#60;1, then x^2&#60;x, x^3&#60;x^2 and so on.  Therefore, <\/p>\n<p>lim<sub>n&#8594;&#8734;<\/sub>x^(n+1)=0, leading to<\/p>\n<p>(1-x^(n+1))\/(1-x)=1\/(1-x)=1+x+x^2+x^3+x^4+&#8230;. when -1&#60;x&#60;1<\/p>\n<p>Above is the basis for the sum of an infinite series whose common ratio, x, follows -1&#60;x&#60;1.<\/p>\n<p>There is even more to say about the identities shown here; I&#8217;ll be following up soon:)<\/p>\n<p>Sources:<\/p>\n<p>Grimaldi, Ralph P.  <u>Discrete and Combinatorial Mathematics<\/u>.  Addison-Wesley:  Toronto, 1994.<\/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 follows the idea of the telescoping sum. In my previous post, I gave an example of a telescoping sum. It implies that 1-x^(n+1)=(1-x)(1+x+x^2+x^3+x^4+&#8230;..+x^n) Dividing both sides by 1-x we arrive at (1-x^(n+1))\/(1-x)=1+x+x^2+x^3+&#8230;..+x^n which is familiar from the sum &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/math-implications-from-the-telescoping-sum\/\"> <span class=\"screen-reader-text\">Math:  implications from the telescoping sum<\/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":[3],"tags":[1020,1022,1021],"class_list":["post-12114","post","type-post","status-publish","format-standard","hentry","category-math","tag-infinite-geometric-series","tag-sum-of-geometric-series","tag-telescoping-sum"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/12114","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=12114"}],"version-history":[{"count":40,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/12114\/revisions"}],"predecessor-version":[{"id":29644,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/12114\/revisions\/29644"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=12114"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=12114"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=12114"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}