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

Question

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

Corben Pittman · Accepted Answer

Given,We have to find the Taylor series for the function f(x)=tan−1⁡(4x) centered at a=0Formula Used:The general term for the Taylor Series f(x)=∑n=0∞fn(a)n!(x−a)n centered at origin a.Calculation:f(x)=∑n=0∞fn(0)n!(x)n=f(0)+xf′(0)+x22!f″(0)+x33!f‴(0)+...Given the function f(x)=tan−1⁡(4x)f(x)=tan−1⁡(4x)⇒f(0)=0f′(x)=41+16x2⇒f′(0)=4f″(x)=−128(1+16x2)2⇒f″(0)=0f‴(x)=−128(1−64x+16x2−1024x3)(1+16x2)3⇒f‴(0)=−128...Hence the Taylor series of the function f(x)=tan−1⁡(4x) centered at 0 thenf(x)=f(0)=xf′(0)+x22!f″(0)+x33!f‴(0)+...tan−1⁡(4x)=0+x(4)+x22!×0+x33!×(−128)+...tan−1⁡(4x)=4x−128x33!+...

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

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

Answered question

Answer & Explanation

New Questions in Calculus 2