   Chapter 11.9, Problem 23E

Chapter
Section
Textbook Problem

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

To determine

To find:

i) The power series representation for the function fx=ln1+x1-x

ii) The graph of f  with several partial sums on same screen. Also want to write what happens as n  increases.

Explanation

1) Concept:

i) The power series expansion of the function 11-x=n=0xn=1+x+x2+x3+     x<1

2) Given:

fx=ln1+x1-x

3) Calculation:

The first step is to express the function fx=ln1+x1-x as sum of a power series.

By using the property of logarithm ln1+x1-x=ln1+x-ln(1-x)

Since,ln1+x=11+xdxand ln1-x=-11-xdx

Therefore,

ln1+x-ln1-x=11+xdx+11-xdx

=11-(-x)dx+11-xdx

=n=0(-xn)dx+n=0xndx

=n=0-1nxndx+n=0xndx

=(1-x+x2-x3+x4-)dx+(1+x+x2+x3+x4+)dx

=(2+2x2+2x4+2x6)dx

=n=02x2ndx

=C+n=02x2n+12n+1

Since, the function at x=0 isf0=ln1+01-0=ln1=0

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