Chapter 11.9, Problem 4PT

Using $\frac{1}{1-x}={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{x}^n}$ for |x| < 1 and differentiation, find a power series for $\frac{1}{{(1+x)}^2}$.

1. a) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{x^{n-1}}{n}}$
2. b) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}-n{x}^{n-1}}$
3. c) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{(-n)}^{n+1}{x}^{n-1}}$
4. d) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}n{(-1)}^{n+1}{x}^{n-1}}$