{"id":5740,"date":"2013-04-18T18:04:57","date_gmt":"2013-04-18T18:04:57","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=5740"},"modified":"2014-12-02T08:27:03","modified_gmt":"2014-12-02T08:27:03","slug":"finite-and-infinite-mathematical-series","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2013\/04\/18\/finite-and-infinite-mathematical-series\/","title":{"rendered":"Finite and Infinite Mathematical Series"},"content":{"rendered":"

Overview:\u00a0 What Is A Mathematical Series?<\/strong><\/p>\n

A mathematical series is the sum of the elements of a mathematical sequence.\u00a0 If the series is finite, the sum will be a finite number.\u00a0 For example, in the finite sequence {1, 2, 3, 4, 5}, the series is the sum of all the terms: 1 + 2 +3 +4 + 5 = 15.<\/p>\n

Partial Sums and Sigma<\/strong><\/p>\n

Partial sums can be used, if the series is very long.\u00a0 The final sum in the above example is easy to find, but it can also be approached by finding partial sums, such as the third partial sum in the series, S3<\/sub>, which is the sum of all the values of the first three numbers in the sequence, 1 +2 +3 or 6.\u00a0 In formula terms, directions to find the sum are given by a Greek letter, sigma or \u2211.<\/p>\n

Infinite Mathematical Series<\/strong><\/p>\n

What is the mathematical sequence is infinite?\u00a0 Finding the sum isn’t as easy as in a finite series, and the directions to find partial sums are very important. \u00a0\u00a0For example, in the \u00a0series {2 + 4 +6 +… + 2n …} , the sum of the first term , S1<\/sub>=2, the second partial sum, S2<\/sub>, is 2 +4, or 6, and the third partial sum, S3<\/sub>, is 2 +4 +6 = 12.\u00a0 The series will go on infinitely, because for every 2n, the nth term can always be one number larger.<\/p>\n

Are All Mathematical Series Infinite?<\/strong><\/p>\n

The sum of many mathematical sequences is an infinitely large number, because n is infinitely large.\u00a0 However, some mathematical sequences are infinite, but n becomes infinitely small, and the sum approaches a limit,\u00a0 For example, suppose the mathematical sequence starts with a square with area 1, and then divide the square into halves.\u00a0 The first rectangle will have an area 1\/2 the size of the original square, the second rectangle will have an area 1\/4 the size, the third will have an area 1\/8, the fourth will have the area 1\/16, and so on.\u00a0 The pattern can be expressed as (1\/2)1<\/sup>, (1\/2)2<\/sup>, (1\/2)3<\/sup>…(1\/2)n<\/sup>…\u00a0 If it is written as a series , then 1\/2 +1\/4 +1\/8 +1\/16 +… = 1, because the original square has area 1 and it was just divided into n rectangles.\u00a0 The infinite series has a finite limit.<\/p>\n

Real-World Applications Of Mathematical Series<\/strong><\/p>\n

Mathematical series have a number of useful applications.\u00a0 For example, determining how much money will be earned with a recurring investment is an application of mathematical series used in banking.\u00a0 There are a number of mathematical series that are used in radio and electronics, as well as in physics and computer science.<\/p>\n

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Overview:\u00a0 What Is A Mathematical Series? A mathematical series is the sum of the elements of a mathematical sequence.\u00a0 If the series is finite, the sum will be a finite number.\u00a0 For example, in the finite sequence {1, 2, 3, 4, 5}, the series is the sum of all the terms: 1 + 2 +3 […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2,14],"tags":[1038,1091,1286,1627],"class_list":["post-5740","post","type-post","status-publish","format-standard","hentry","category-algebra","category-pre-calculus","tag-limits","tag-mathematical-series","tag-partial-sums","tag-sigma"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/5740","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=5740"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/5740\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=5740"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=5740"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=5740"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}