   Chapter 11.9, Problem 9E

Chapter
Section
Textbook Problem

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

To determine

To find:

The power series representation and determine the interval of convergence for the function

fx=x-1x+2

Explanation

1) Concept:

The series

11-x=1+x+x2+x3+=n=0xn

Converges when x<1

2) Given:

fx=x-1x+2

3) Calculation:

Given the function

fx=x-1x+2

To represent the function as a sum of a power series,

x-1 can also be written as x+2-3

Therefore, the function becomes

fx=x+2-3x+2

fx=x+2x+2-3x+2

fx=1-3x+2

fx=1-32+x

Divide the numerator and the denominator by 2,

fx=1-321+x2

Which can also be rewritten as

fx=1-321--x2

Comparing this function f(x) with

11-x=1+x+x2+x3+=n=0xnx<1

gives,

fx=1-321--x2= 1-n=032·-x2n

Which can also be written as

1-n=0 <

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