   Chapter 11.9, Problem 19E

Chapter
Section
Textbook Problem

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

To determine

To find:

i) The power series representation using for the function

fx=1+x1-x2

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

f'x=c1+2c2x-a+3c3x-a2=n=1ncnx-an-1

then the radius of convergence of the power series f'x=n=1ncnx-an-1 has the radius of convergence R.

2) Given:

fx=1+x1-x2

3) Calculation:

Given the function

fx=1+x1-x2

This can also be rewritten as

fx=11-x2+x1-x2

fx=11-x2+11-x2·x

We know that

11-x2= n=0(n+1)xn

Thus,

fx=1+x1-x2= n=0(n+1)xn+ n=0(n+1

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