Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation. f(x)=tan^{-1}4x,a=0

DofotheroU

DofotheroU

Answered question

2021-01-02

Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
f(x)=tan14x,a=0

Answer & Explanation

Corben Pittman

Corben Pittman

Skilled2021-01-03Added 83 answers

Given,
We have to find the Taylor series for the function f(x)=tan1(4x) centered at a=0
Formula Used:
The general term for the Taylor Series f(x)=n=0fn(a)n!(xa)n centered at origin a.
Calculation:f(x)=n=0fn(0)n!(x)n
=f(0)+xf(0)+x22!f(0)+x33!f(0)+...
Given the function f(x)=tan1(4x)
f(x)=tan1(4x)f(0)=0
f(x)=41+16x2f(0)=4
f(x)=128(1+16x2)2f(0)=0
f(x)=128(164x+16x21024x3)(1+16x2)3f(0)=128
...
Hence the Taylor series of the function f(x)=tan1(4x) centered at 0 then
f(x)=f(0)=xf(0)+x22!f(0)+x33!f(0)+...
tan1(4x)=0+x(4)+x22!×0+x33!×(128)+...
tan1(4x)=4x128x33!+...
Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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