Proof of an infinite series formula If $$|x| < 1$$ Prove that
$$\begin{align}\large 1 + 2x + 3x^2 + 4x^3 + \dots = \frac{1}{(1 - x)^2}\end{align}$$

Another approach:
As @Andre said, you have $$S=\sum_{i=0}^\infty x^i=\frac1{1-x}.$$ You wanted to calculate $$T=\sum_{i=1}^\infty ix^{i-1}.$$
Final job: Consider the difference $$xT-S$$ and use the explicit form of $S$.
