Singularity of Geometric-like series

Question

I'm trying to figure out exactly what happens as $r\to 1$ of the following function: $$ f(r) = \sum_{n>0} n^s r^n, $$ where the domain of $f$ is the positive real numbers less than $1$, and $s$ is some positive real number.

For the case where $s$ is a positive integer, one can exactly write down the closed form of $f$, and one finds that the smallest value of $k$ for which $f(r)\cdot (1-r)^{(k)}$ is a bounded function is $k=s+1$. (Please forgive me if I'm off by $1$.)

1. Is there some kind of a "closed form" expression for $f$ when $s$ is a rational number? A real number? A complex number?

2. Is "$k=s+1$" the smallest number for which $f(r) \cdot (1-r)^k$ is bounded even when $s$ is not a positive integer?

It would be great if someone could point me to a reference rather than spoon feeding me the answer, but I don't mind the spoon feeding :)

Thanks in advance!

Answer 1

Your series is the polylogarithm function (up to the sign of s) See e.g. http://en.wikipedia.org/wiki/Polylogarithm and references there.