   Chapter 11.9, Problem 18E

Chapter
Section
Textbook Problem

# Find a power series representation for the function and determine the radius of convergence.18. f ( x ) = ( x 2 − x ) 3

To determine

To find: The power series representation for the function f(x)=(x2x)3 and determine the radius of convergence of the series.

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 for 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)=(x2x)3

Calculation:

Consider result (2) the power series f(x)=(x2x)3 is 11x=n=0xn

Replace x for x2 .

11(x2)=n=0(x2)n22x=n=0(12)nxn

Divide both sides by 2.

12x=n=0(12)n+1xn

Differentiate both sides as shown below,

ddx[12x]=ddx[n=0(12)n+1xn]1(2x)2=n=0(12)n+1ddxxn1(2x)2=n=1(12)n+1(n)xn1

The summation part is starting from n=1 , because the first term eliminated during differentiation.

Differentiate both sides.

ddx[1(2x)2]=ddx[n=1(12)n+1nxn1]2(2x)3=n=1(12)n+1nddxxn12(2x)3=n=1(12)n+1n(n1)xn2

In the summation part is starting from n=2 after differentiation, because the derivative of the first term was zero

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