Chapter 9.9, Problem 19E

Using a Power Series In Exercises 19-28, use the power series

$\frac{1}{1+x}={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{(-1)}^n{x}^n},\left|x\right|<1$

to find a power series for the function, centered at 0, and determine the Interval of convergence.

$f(x)=-\frac{1}{{(x+1)}^2}-\frac{d}{dx}\left[\frac{1}{x+1}\right]$