Chapter 6.2, Problem 94E

In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series.

94. Evaluate ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{(-1)}^n}{1+n}}$ as ${\displaystyle {\int }_0^{1}f(t)dt}$where $f(x)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{(-1)}^n{x}^{2n}=\frac{1}{1+x^2}.}$