   Chapter 11.9, Problem 22E

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 4 )

To determine

To find:

i) The power series representation for the function fx=ln(1+x4)

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

Solution:

i) The power series representation of the function is

ln1+x4 =n=1-1n-1x4nn=x4-x22+x123-x164+x205-

ii) The graph of f  with several partial sums on same screen. From the graph, as n increases the partial sum sn(x) approximates the function to its interval of convergence [-1, 1]

Explanation

1) Concept:

i) The power series expansion of the function ln1+x =n=1-1n-1xnnx<1

2) Given:

fx=ln(1+x4)

3) Calculation:

The first step is to express the function ln(1+x4) as sum of a power series.

By using the power series of the of the function ln1+x, replace x by x4

ln1+x =n=1-1n-1xnn

ln1+x4 =n=1-1n-1x4nn

The expansion of power series

n=1-1n-1x4nn=x4-x22+x123-x164+x205-

This series convergent when x4<1 i.e

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