Question

# Starting with the geometric series \sum_{n=0}^\infty x^n, find the sum of the series\sum_{n=1}^\infty nx^{n-1},\ |x|<1

Series

Starting with the geometric series $$\sum_{n=0}^\infty x^n$$, find the sum of the series
$$\sum_{n=1}^\infty nx^{n-1},\ |x|<1$$

2021-06-14

Consider the geometric series,
$$\sum_{n=0}^\infty x^n=1+x+x^2+x^3+...$$
$$=\frac{1}{1-x}$$
Find the sum of the series $$\sum_{n=1}^\infty nx^{n-1},\ |x|<1$$
$$\sum_{n=1}^\infty nx^{n-1}=1+2x+3x^2+4x^3+...$$
$$=(1+x+x^2+x^3+...)+(x+2x^2+3x^2+4x^3+...)$$
$$=\frac{1}{1-x}+x(1+2x+3x^2+4x^3+...)$$
$$=\frac{1}{1-x}+x\sum_{n=1}^\infty nx^{n-1}$$
$$\Rightarrow\sum_{n=1}^\infty nx^{n-1}-x\sum_{n=1}^\infty nx^{n-1}=\frac{1}{1-x}$$
$$(1-x)\sum_{n=1}^\infty nx^{n-1}=\frac{1}{1-x}$$
$$\sum_{n=1}^\infty nx^{n-1}=\frac{1}{(1-x)^2}$$