Chapter 11.2, Problem 3QC

Use the result of Example 4 to write a series representation for $\ln \frac{1}{2}=-\ln 2$.

Example 4 Differentiating and Integrating Power Series

Consider the geometric series

$f(x)=\frac{1}{1-x}={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{x}^k=1+x+x^2+x^3+\cdots ,\text{ for }\left|x\right|<1.}$

1. a. Differentiate this series term by term to find the power series for f′ and identify the function it represents.
2. b. Integrate this series term by term and identify the function it represents.