{"id":1722,"date":"2020-08-15T07:35:06","date_gmt":"2020-08-15T11:35:06","guid":{"rendered":"http:\/\/josmfs.net\/?p=1722"},"modified":"2024-09-06T16:58:13","modified_gmt":"2024-09-06T20:58:13","slug":"serious-series","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2020\/08\/15\/serious-series\/","title":{"rendered":"Serious Series"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-1720\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/08\/Serious-Series-Fig.jpg\" alt=\"\" width=\"202\" height=\"183\" \/>The following problem comes from a 1961 exam set collected by Ed Barbeau of the University of Toronto.\u00a0 The discontinued exams (by 2003) were for 5<sup>th<\/sup> year Ontario high school students seeking entrance and scholarships for the second year at a university.<\/p>\n<p>\u201cIf <em>s<sub>n<\/sub><\/em> denotes the sum of the first <em>n<\/em> natural numbers, find the sum of the infinite series<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1723 aligncenter\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/08\/Serious-Series.jpg\" alt=\"\" width=\"142\" height=\"44\" \/>.\u201d<\/p>\n<p>Unfortunately, the \u201cGrade XIII\u201d exam problem sets were not provided with answers, so I have no confirmation for my result.\u00a0 There may be a cunning way to manipulate the series to get a solution, but I could not see it off-hand.\u00a0 So I employed my tried and true power series approach to get my answer.\u00a0 It turned out to be power series manipulations on steroids, so there must be a simpler solution that does not use calculus.\u00a0 I assume the exams were timed exams, so I am not sure how a harried student could come up with a quick solution.\u00a0 I would appreciate any insights into this.<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Serious-Series-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/08\/Serious-Series-191217.pdf\">Serious Series<\/a> for a solution.<\/p>\n<p>(<strong>Update 1\/18\/2021, 9\/6\/2024<\/strong>) <strong>Other Solutions<\/strong><br \/>\n<!--more--><\/p>\n<p>(<strong>Update 1\/18\/2021) <\/strong>I received another <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/01\/Serious_Deret_Evan.pdf\">solution<\/a> from Evander Tandiarrang (of <a href=\"https:\/\/josmfs.net\/2020\/03\/14\/hard-geometric-problem\/\">Hard Geometric Problem<\/a> fame) from Papua New Guinea, now in 7<sup>th<\/sup> grade!\u00a0\u00a0 It looks correct to me and only relies on the geometric series without explicit calculus.<\/p>\n<p>Needless to say, I was nowhere near such mathematical prowess in 7<sup>th<\/sup> grade.<\/p>\n<p style=\"padding-left: 40px;\"><em>Pada Senin, 18 Januari 2021 11.06.52 GMT+9 <\/em><\/p>\n<p style=\"padding-left: 40px;\"><em>Good morning and Happy New Year, <\/em><\/p>\n<p style=\"padding-left: 40px;\"><em>Every time I practice with mathematic and science problems, and now I try to solved the Serious Series problem, but may be not correct.\u00a0 Please give me a correction for my solution.<\/em><\/p>\n<p style=\"padding-left: 40px;\"><em>Regards,<\/em><\/p>\n<p style=\"padding-left: 40px;\"><em>Evander T.<\/em><\/p>\n<p>(<strong>Update 9\/6\/2024) <\/strong>I received a belated <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/09\/Serious-Series-Francois-Mignard.pdf\">solution<\/a> from Fran\u00e7ois Mignard that also avoids calculus.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The following problem comes from a 1961 exam set collected by Ed Barbeau of the University of Toronto.\u00a0 The discontinued exams (by 2003) were for 5th year Ontario high school students seeking entrance and scholarships for the second year at a university. \u201cIf sn denotes the sum of the first n natural numbers, find the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[32,205,152,230],"class_list":["post-1722","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-calculus","tag-ed-barbeau","tag-power-series","tag-sequences-and-series"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1722","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=1722"}],"version-history":[{"count":7,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1722\/revisions"}],"predecessor-version":[{"id":4062,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1722\/revisions\/4062"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1722"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1722"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1722"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}