Simple question but I'm getting lost along the way: Find the infinite series representation of
$$\frac{1}{(x-1)^3}$$
Looks suspiciously like power series; it can be rewritten as: $$\frac{-1}{2}\cdot\frac{d^2}{dx^2}\frac{1}{1-x}$$
How can this be translated to a power series now that $\frac{1}{1-x}$ has been isolated? From looking up the answer it is:
$$-\sum_{n=0}^\infty\frac{(n+1)(n+2)x^n}{2}$$
But it's unclear to me where $(n+1)(n+2)$ comes from. How are functions in the form $\frac{1}{(1-x)^n}$ solved?
Remember that $$\frac{1}{1-x}=\sum_{n=0}^{\infty}{x^n},\quad|x|<1.$$ Now take the second derivative on both sides and you'll see where the $(n+1)(n+2)$ comes from (think about why you can take the derivative here). You can apply this technique to any function of the form $$\frac{1}{(1-x)^n}.$$