Chapter 11.9, Problem 40E

(a) Starting with the geometric series ${\displaystyle {\sum }_{n=0}^{\infty }{x}^n}$, find the sum of the series

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}n{x}^{n-1}}\left|x\right|<1$

(b) Find the sum of each of the following series,

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

(ii) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{n}{2^n}}$

(c) Find the sum of each of the following series.

(i) ${\displaystyle \underset{n=2}{\overset{\infty }{\sum }}n\left(n-1\right)x^n},\left|x\right|<1$

(ii) ${\displaystyle \underset{n=2}{\overset{\infty }{\sum }}\frac{n^2-n}{2^n}}$

(iii) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{n^2}{2^n}}$