{"id":3754,"date":"2021-07-12T21:14:48","date_gmt":"2021-07-12T15:44:48","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=3754"},"modified":"2026-04-13T05:09:13","modified_gmt":"2026-04-12T23:39:13","slug":"what-is-maclaurin-series","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-maclaurin-series\/","title":{"rendered":"What is Maclaurin series?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overvie<\/strong>w<\/h2>\n\n\n\n<p>A Maclaurin series is a special case of Taylor series. The Taylor\u2019s series is an expansion of a particular function about a point \u2018a\u2019.<\/p>\n\n\n\n<p>It contains infinite sum of terms, and each exponent has a larger exponent.<\/p>\n\n\n\n<p>The Taylor\u2019s is-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/UHFLIc2_Bjqe6QFmo6rbuiU_i_oB_eQpkNZup83dwTAW1qBTQvUlN20Y79FdEIQcHcKPLKOX_pztPSuC_culo7zeMaDBVjan7lZk8e_60ekx5Dc3DmfQuAOdTOgiPtuufwfdKdOS\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Maclauin\u2019s series-<\/p>\n\n\n\n<p>When we put a = 0 in Taylor\u2019s formula, it reduces to-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/EguESUZWrhsPwmHNtj8GS-neCxxNU9UZzKfMj8-Z9PuoT5SOPP1lWXGmYj7Wv3x-NT1H0ZkgO6uLTHZ8MmVGPFYs7Z7fRC66FiW3-KxloHxFajOi2JXR_D-iFf2gYGq9SbAA3VTy\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Which is the Maclaurin&#8217;s series.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Taylor and Maclaurin Series\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/a58otkmkHtk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Taylor&#8217;s series<\/h2>\n\n\n\n<p>Taylor\u2019s series is considered as an expansion of a function into an infinite sum of terms.<\/p>\n\n\n\n<p>The formula was introduced by Brook Taylor in 1715, after that the series named as Taylor\u2019s series.<\/p>\n\n\n\n<p>To get the approximate value of the function, first few terms can be used.<\/p>\n\n\n\n<p>Taylor\u2019s series has applications in modern physics, classical physics, mathematics, advanced mathematics etc.<\/p>\n\n\n\n<p><strong>Definition<\/strong><\/p>\n\n\n\n<p>The Taylor\u2019s series for the function f(x) about x = a is be defined as-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/HcXvX3R8glQ8Lh4YDcOAqf2CMpjkrGeV2yQ-_wI4K_XhWt-5G8azRnUbAE1vYSvBoWA-uA9j7FNH6hdchH76tT3Z2mSKsfq6BYB4pS28NzOwXyMNYEKvD3DS0IhJH3HpcTkyakm1\" alt=\"\"\/><\/figure>\n\n\n\n<p>OR<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/-0Vebd3ag-9CKIwPxUf10YhIjUA7OR4wx1mVr_djD8dkJcBN11jaxSniAh-jsbc64nSTdwDiUGkrpOg169W_xtP3KARLQaVIJHYnENFiadZh95-DwE_YNuX6TJ3h8kg8IF-IcSve\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Note- <\/strong>If we put x = 0 then the series we get is called Maclaurin\u2019s series for f(x)<\/p>\n\n\n\n<p><strong>Example: For the function f(x)&nbsp; = cos x. find the Taylor\u2019s series about x = 0.<a href=\"https:\/\/bgd.cfga.gov.mn\/\" style=\"color: #ffffff; text-decoration: none;\">https:\/\/bgd.cfga.gov.mn\/<a><\/strong><\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>First we will find some derivatives of the function f(x), then we find the value of of these at x = 0.<\/p>\n\n\n\n<p>f(x) = cos x&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; f(0) = 1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/p>\n\n\n\n<p>f\u2019(x)&nbsp; = -sin x &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; f\u2019(0) = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/p>\n\n\n\n<p>f\u2019\u2019(x)&nbsp; = -cos x &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; f\u2019\u2019(0) = -1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/p>\n\n\n\n<p>f\u2019\u2019\u2019(x)&nbsp; = sin x&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; f\u2019\u2019\u2019(0) = 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<\/p>\n\n\n\n<p>f\u2019\u2019\u2019\u2019(x)&nbsp; = cos x&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; f\u2019\u2019\u2019\u2019(0) = 1<\/p>\n\n\n\n<p>So that putting these values in Taylor\u2019s series, we get-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/LfNcKotq86QMXjSbG7U0lIzQsnevQfP5MLzD6rkGpYofR_MD3Rh8AQlxF0dkGxWHCQceCwmi03weYy4vzYqMGTe_ugGQFk8ixjpTzeMc2fVWFPwFct1muJUAJr0hHMQwppA7TPT7\" alt=\"\"\/><\/figure>\n\n\n\n<p>Hence we get the series for cos x about x = 0&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/IHFFnjXNeCeTlcyXB3Soa3nAb4c77GsVZDexZq3qlrYt2xAeOkDFW5fFE-RwO7d5x880d8zSreEiblSKov5WLVlFTRpO9oJ5kqBsZnsMH51xQcGVJV0qb_LfuUPLUPXQAPneya0n\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Solved examples<\/strong> of Maclaurin series<\/h2>\n\n\n\n<p><strong>Example: By using Maclaurin\u2019s series expand tan x.<\/strong><\/p>\n\n\n\n<p>Let-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/nhbkTjoB621QCe9ZE0nJZjkcYxKZdHhKrPlinlgPdQBujL5Z3WfLoC4N53aZEpTbXj0x5Z-r-8_pYkfCl0hedNXU-dqROYNl8ZYEZOFyBBNM8pKYw61rJwEPw3qj9jTYY5_L2M1D\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>So that putting these values in Maclaurin\u2019s series we get-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/ASaQ-gHIgIfZNg4tKj1hHbKqWsWEANfb1kHwQD1YFou9C5Fl2twr0tF6rPRTIlGE6IMtFo8LD3KWyc1935AVXLuuu6vsuHlRhIVeJr43CjPb5GuK8GjlNMnAt1Dq0rRi4fU5rqRI\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Example: Expand &nbsp;<img loading=\"lazy\" decoding=\"async\" width=\"38\" height=\"22\" src=\"https:\/\/lh3.googleusercontent.com\/GhPayS9xKz2NtBuBC43EWIx1SEkjhDpCcHvZ278H33wmyJe_j33-Gc5b2ggKtCjM6RJ70bPLc4tM60jfpuRBWVgLEK_6eww3Q6cZz57x3E-Z_-B7zUJq33hMB3Fo4jipojn5B-GC\"> by using Maclaurin\u2019s series.<\/strong><\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Let<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/F0YBj-3WGxfx3OscKIkVSg2cZKDdPIFYv0NMZqm8jy79FJIBBS2dO14fj3vqbQ_fMfTVkHLuab0pFk5Sb8LjCEChcjHWKY4TZ4CNbEam5WWQp1Yej-qKnNnP7Kr05BLFHfabSB-N\" alt=\"\"\/><\/figure>\n\n\n\n<p>So that putting these values in Maclaurin\u2019s series, we get-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/OilCmaEpLPFTUtIW-XWg3th0az4k7Gjm0Cux23fBZJBCPJMprkXeq7bQcQ_YlPaL9W9VdSZhddn2vvk2XVjWUBUkIi_xTNtz190miUtlKxzqFjEbkfTP5OGdsO8EDk5urD1Z3xhc\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Or<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/ss52_iKsHsm0-yRrEP-KHCzhXomVru3TF2dGTHGs-_MGHi2jvkh6Wyf6nFRGoOjDs4efa0-vWL_Xs_hYaSkA2mFVuuMNC5L4N2xnatTHVoxPMV61qDU2RygL62xNptRPJfNN4ClR\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Interested in learning about similar topics? Here are a few hand-picked blogs for you! <a href=\"https:\/\/dafm.stamforduniversity.edu.bd\/\">https:\/\/dafm.stamforduniversity.edu.bd\/<\/a><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-taylors-series\" target=\"_blank\" rel=\"noreferrer noopener\">What is Taylor&#8217;s series?<\/a> <\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Maclaurin series is a special case of Taylor series. The Taylor\u2019s series is an expansion of a particular function about a point \u2018a\u2019. <\/p>\n","protected":false},"author":3,"featured_media":4078,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-3754","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is Maclaurin series? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"A Maclaurin series is a special case of Taylor series. 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