   Chapter 11.9, Problem 14E

Chapter
Section
Textbook Problem

(a) Use Equation 1 to find a power series representation for f ( x ) = ln ( 1 − x ) . What is the radius of convergence?(b) Use part (a) to find a power series for f ( x ) = x ln ( 1 − x ) .(c) By putting x = 1 2 in your result from part (a), express ln 2 as the sum of an infinite series.

To determine

(a)

To find:

i) The power series representation using Equation 1 for the function

fx=ln(1-x)

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

fxdx=C+c0x-a+c1x-a22+c2x-a33+=C+n=0x-an+1n+1

has the radius of convergence R.

2) Given:

fx=ln(1-x)

3) Calculation:

Given the function

fx=ln(1-x)

We know that,

11-x dx= -ln1-x+C

And

11-x dx= 1+x+x2+dx

=x+x22+x33++C

=n=1xnn+C for  x <1

So,

-

To determine

(b):

To find:

The power series for fx=x ln(1-x)  using part (a)

To determine

(c):

To express:

ln2 as the sum of an infinite series, by putting x=12 in the result of part (a)

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