Identifying a function from its power series representation What functions are represented by the following power series?
$$\sum\limits_{k=1}^{\infty}kz^k \quad \quad \quad \sum\limits_{k=1}^{\infty}k^2z^k$$
Would this involve using a Taylor expansion? I cannot think of any general functions that follow a similar representation (ie. $e^x$ involves factorials).
I wonder if I need to be considering a differentiation of a general $\sum\limits_{k=1}^{\infty}z^k$, since the two functions seem to follow that format? However the indices don't tally, as a differentiation of the general term above would be $\sum\limits_{k=1}^{\infty}kz^{k-1}$.

Let $f(z)=\frac1{1-z}=\sum_{k=0}^\infty z^k$, then 
$$\sum_{n=1}^\infty kz^k=z\sum_{n=1}^\infty kz^{k-1}=zf'(z).$$
$$f''(z)=\sum_{k=2}^\infty k(k-1)z^{k-2}=\sum_{k=2}^\infty k^2z^{k-2}-\sum_{k=2}^\infty kz^{k-2}.$$
So, 
$$\sum_{k=2}^\infty k^2z^{k-2}=f''(z)+\frac{f'(z)}{z},$$
and
$$\sum_{k=2}^\infty k^2z^k=z^2f''(z)+zf'(z).$$
