{"id":1675,"date":"2021-05-26T12:03:11","date_gmt":"2021-05-26T06:33:11","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=1675"},"modified":"2021-10-30T07:40:43","modified_gmt":"2021-10-30T07:40:43","slug":"what-is-taylors-series","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-taylors-series\/","title":{"rendered":"What is Taylor’s series?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

Taylor\u2019s series is considered as an expansion of a function into an infinite sum of terms.<\/p>\n\n\n\n

The formula was introduced by Brook Taylor in 1715, after that the series named as Taylor\u2019s series.<\/p>\n\n\n\n

To get the approximate value of the function, first few terms can be used.<\/p>\n\n\n\n

Taylor\u2019s series has applications in modern physics, classical physics, mathematics, advanced mathematics etc.<\/p>\n\n\n\n

Definition<\/strong><\/h2>\n\n\n\n

The Taylor\u2019s series for the function f(x) about x = a is be defined as-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

OR<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

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

Let\u2019s do an  example to get the concept-<\/p>\n\n\n\n

Example: For the function f(x)  = cos x. find the Taylor\u2019s series about x = 0.<\/strong><\/p>\n\n\n\n

Solution:<\/p>\n\n\n\n

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

f(x) = cos x                                                            f(0) = 1                      <\/p>\n\n\n\n

f\u2019(x)  = -sin x                                                         f\u2019(0) = 0                     <\/p>\n\n\n\n

f\u2019\u2019(x)  = -cos x                                                       f\u2019\u2019(0) = -1                      <\/p>\n\n\n\n

f\u2019\u2019\u2019(x)  = sin x                                                        f\u2019\u2019\u2019(0) = 0                    <\/p>\n\n\n\n

f\u2019\u2019\u2019\u2019(x)  = cos x                                                      f\u2019\u2019\u2019\u2019(0) = 1<\/p>\n\n\n\n

Putting these values in Taylor\u2019s series, we get-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence we get the series for cos x about x = 0 <\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

\u00a0Similarly we can get the expansion of other functions by using the Taylor\u2019s series.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n