Proof of sum of infinite series including squared variable times a geometric series

Question

I'm wondering how to prove the following series:

$$\alpha^2 \sum_{n=1}^\infty n^2 (1-\alpha)^{(n-1)} = \frac{2-\alpha}{\alpha}$$

Answer 1

The idea is to use that for $|x|<1$

$$\frac{d}{dx}\sum_{n=0}^\infty x^n=\sum_{n=1}^\infty nx^{n-1}$$

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

and some manipulation for geometric series.