Overview
Taylor’s series is considered as an expansion of a function into an infinite sum of terms.
The formula was introduced by Brook Taylor in 1715, after that the series named as Taylor’s series.
To get the approximate value of the function, first few terms can be used.
Taylor’s series has applications in modern physics, classical physics, mathematics, advanced mathematics etc.
Definition
The Taylor’s series for the function f(x) about x = a is be defined as-
OR
Note- If we put x = 0 then the series we get is called Maclaurin’s series for f(x)
Let’s do an example to get the concept-
Example: For the function f(x) = cos x. find the Taylor’s series about x = 0.
Solution:
First we will find some derivatives of the function f(x), then we find the value of of these at x = 0.
f(x) = cos x f(0) = 1
f’(x) = -sin x f’(0) = 0
f’’(x) = -cos x f’’(0) = -1
f’’’(x) = sin x f’’’(0) = 0
f’’’’(x) = cos x f’’’’(0) = 1
Putting these values in Taylor’s series, we get-
Hence we get the series for cos x about x = 0
Similarly we can get the expansion of other functions by using the Taylor’s series.
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