   Chapter 11.9, Problem 23E

Chapter
Section
Textbook Problem

# Find a power series representation for f, and graph f and several partial sums sn(x) on the same screen. What happens as n increases?23. f ( x ) = ln ( 1 + x 1 − x )

To determine
The power series representation for the function f(x)=ln(1+x1x) and graph f(x) and several partial sums sn(x) .
Explanation

Given:

Let f(x)=ln(1+x1x)

Result used:

(1) “The power series representation of 11x is n=0xn .”

Calculation:

Let, f(x)=ln(1+x1x) , (1)

Expand the expression and find the derivative of f(x) .

f(x)=ln(1+x1x)=ln(1+x)ln(1x)f(x)=11+x+11x=11(x)+11x

By using the result (1) f(x) is shown below,

f(x)=n=0((x)n+xn)=n=0((1)nxn+xn)=n=0((1)n+1)xn

Integrate on both sides to get f(x) .

f(x)=n=0((1)n+1)xn=n=0((1)n+1)xn+1n+1+C

Set x=0 in (1) to determine the value of C.

f(0)=ln(1+(0)1(0))=ln(11)=ln(1)=0

The summation part is zero when x=0 , therefore C=0

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