   Chapter 11.9, Problem 16E

Chapter
Section
Textbook Problem

Find a power series representation for the function and determine the radius of convergence. f ( x ) = x 2 tan − 1 ( x 3 )

To determine

To find:

i) The power series representation using for the function

fx=x2tan-1(x3)

ii) The radius of convergence.

Explanation

1) Concept:

If the power series cnx-an has radius of convergence R>0, then the function f defined by

fx=c0+c1x-1+c2x-12+=n=0cnx-an is differentiable (and therefore continuous) on the interval (a-R, a+R) and

fxdx=C+c0x-a+c1x-a22+c2x-a33+=C+n=0x-an+1n+1

has the radius of convergence R.

2) Given:

fx=x2tan-1(x3)

3) Calculation:

Given the function

fx=x2tan-1(x3)

We know that

ddxtan-1 x=1(1+x2)

Therefore, with the help of the chain rule,

ddxtan-1x3=3x21+x6

Since,

tan-1x= 11+x2 dx= n=0-1nx2n+12n+1

Integrating the power series for 3x21+x6 gives,

tan-1x3=3x21+x6 dx

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