How to use a generating function to work out an infinite sum

Question

I have the infinite sums:

$$\sum_{k=0}^\infty k^2a^k \quad \text{and}\quad \sum_{k=0}^\infty ka^k$$

where, $\left\lvert a \right\rvert<1$. I was able to find the answers to the infite sums here, but I am interested on how I could use generating functions to get the same answer.

I am unsure of how to even begin this problem. Thanks for any help in advance.

Answer 1

Let's consider the geometric series \begin{align*} \sum_{k=0}^{\infty}x^k=\frac{1}{1-x} \end{align*}

Differentiation and multiplication with $x$ results in \begin{align*} \left(xD_x\right)\sum_{k=0}^{\infty}x^k=x\sum_{k=0}^\infty kx^{k-1}=\sum_{k=0}^\infty kx^k \end{align*}

We conclude by applying the operator $(xD_x)$ once and twice \begin{align*} \sum_{k=0}^{\infty}kx^k&=(xD_x)\frac{1}{1-x}=\ldots=\frac{x}{(1-x)^2}\\ \sum_{k=0}^{\infty}k^2x^k&=(xD_x)^2\frac{1}{1-x}=\ldots=\frac{x(x+1)}{(x-1)^3} \end{align*}