Explain why $\sum\limits_{k=0}^\infty k^2\cdot x^k=-\frac{x(x+1)}{(x-1)^3}$

Question

I was learning generating functions and met this summation. I used maple and it gave $-\frac{x(x+1)}{(x-1)^3}$, but how does it get here? I've forgotten most of the knowledge about series. Does anyone can tell me the intermediate steps?

Answer 1

$$\sum_{k=0}^{\infty}k^2x^k=x\cdot\frac{\mathrm d}{\mathrm dx}\left(x\cdot\frac{\mathrm d}{\mathrm dx}\sum_{k=0}^{\infty}x^k\right)$$

Answer 2

Note that $k^2 = (k+1)^2-2 k - 1$ and that $x\mapsto kx^k$ looks very much like the derivative of $x\mapsto x^k$. I suppose that this will help you to continue.

The result you had from Maple is correct.