Challenge Level

The sum of the infinite geometric series $1 + x + x^2 + x^3 + \cdots$ and the binomial series are well known. How are the two related?

Show that $$\sum_{n=0}^\infty n x^n = {x\over(1-x)^2}$$ and find $$\sum_{n=0}^\infty n^2x^n.$$ Outline a method for finding $$\sum_{n=0}^\infty n^kx^n$$ where you do not have to carry out this computation beyond $k=2$.

Experiment with other expansions to try to find out the values for other interesting series.