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 \(\sum_{n=1}^\infty nx^{n-1},\ |x|<1\)

\(\sum_{n=1}^\infty nx^{n-1}=1+2x+3x^2+4x^3+...\)

\(=(1+x+x^2+x^3+...)+(x+2x^2+3x^2+4x^3+...)\)

\(=\frac{1}{1-x}+x(1+2x+3x^2+4x^3+...)\)

\(=\frac{1}{1-x}+x\sum_{n=1}^\infty nx^{n-1}\)

\(\Rightarrow\sum_{n=1}^\infty nx^{n-1}-x\sum_{n=1}^\infty nx^{n-1}=\frac{1}{1-x}\)

\((1-x)\sum_{n=1}^\infty nx^{n-1}=\frac{1}{1-x}\)

\(\sum_{n=1}^\infty nx^{n-1}=\frac{1}{(1-x)^2}\)