   Chapter 11.9, Problem 16E

Chapter
Section
Textbook Problem

# Find a power series representation for the function and determine the radius of convergence.16. f(x) = x2 tan−1(x3)

To determine

To find: The power series representation for the function f(x)=x2tan1(x3) and determine the radius of convergence.

Explanation

Result used:

(1) Ratio test:

If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent.

(2) The power series representation of 11x=n=0xn .

(3) “The sum of the geometric series with initial term a and common ratio r is n=0arn=a1r .”

Given:

The function is f(x)=x2tan1(x3) .

Calculation:

Obtain the derivative of tan1(x3) .

Let u=x3 .

ddx[tan1(x3)]=ddx[tan1(u)]=ddu[tan1(u)]dudx=11+u2dudx

Substitute u=x3 ,

11+u2dudx=11+(x3)2ddx(x3)=11+x63x2=3x21(x6)

This is geometric sum with a=3x2 and r=x6 by the result (3).

3x211(x6)=n=03x2(x6)n=n=0(1)n(x6)n3x2=n=0(1)n3x6n+2=3x23x8+3x143x20+

That is, ddx[tan1(x3)]=3x23x8+3x143x20+

Integrate on both sides,

ddx[tan1(x3)]dx=(3x23x8+3x143x20+)dxtan1(x3)=((3x33)39x9+315x15321x21+)+C

tan1(x3)=x339x9+315x15321x21++C (1)

Substitute x=0 and obtain the value C.

tan1(0)=0+CC=0       ( tan1(0)=0)

Thus, the value of C=0

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