Convergence of $\sum_{n=0}^{\infty}ne^{-\beta n}$ I am working in a problem of statistical mechanics and a partition function I found is of the form:
$$
Z = \sum_{n=0}^{\infty}(n+1)e^{-\beta n}
$$
I used the ratio test and the series converges. I want to find a closed form for this. I tried:
$$
\sum_{n=0}^{\infty}(n+1)e^{-\beta n} = 
\sum_{n=0}^{\infty}ne^{-\beta n} + 
\sum_{n=0}^{\infty}e^{-\beta n}
$$
The second term of the right-hand side is simply $1/(1-e^{\beta})$ (geometric series). So the question comes down to: does this series have a closed form?
$$
\sum_{n=0}^{\infty}ne^{-\beta n}
$$
where $\beta>0$

Hint
Yes. Denote $f(\beta)$ the geometric series 
$$f(\beta)=\sum_{n=0}^\infty e^{-\beta n}$$
what's $f'(\beta)$? Can you take it from here?
