   Chapter 11.9, Problem 20E

Chapter
Section
Textbook Problem

# Find a power series representation for the function and determine the radius of convergence.20. f ( x ) = x 2 + x ( 1 − x ) 3

To determine

To find: The power series representation for the function for the function f(x)=x2+x(1x)3 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 is 11x=n=0xn

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

Given:

The series is f(x)=x2+x(1x)3

Calculation:

Let f(x)=x2+x(1x)3 .

Then, by the result (2) the power series is 11x=n=0xn .

Double differentiate for 11x=n=0xn on both sides with respect to x

dydx=ddx[11x]dydx=dd(1x)[11x]×d(1x)ddydx=1(1x)2×(1)dydx=1(1x)2

Differentiate the series as shown below:

dydx=(x1)2d2ydx2=2(x1)3=2(x1)3=2(1x)3

Therefore, 2(1x)3=n=2n(n1)xn2 .

Multiply both sides by x2+x2 for simplification.

2(x2+x)2(1x)3=n=2n(n1)xn2×x2+x2(x2+x)(1x)3=n=2n(n1)2((x2+x)xn2)=n=2n(n1)2(xn+xn1)(x2+x)(1x)3=n=2n(n1)2xn+n=2n(n1)2xn1

Replace n by n1 in the second part of the summation

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