How can I evaluate this infinite exponential series using a method I can learn?

[ShayanJ]

I've just encountered the infinite series $\sum_{n=0}^\infty n e^{-n\lambda}$. I know that in general its not possible to evaluate an infinite series but because wolframalpha.com could evaluate it(which gave the result $\frac{e^{\lambda}}{(e^\lambda-1)^2}$), it seems to me that this one has can be solved by a method that I can learn. So I want to learn it. Any ideas how to do it?
Thanks

 

[Svein]

Here are the elements of a solution. Some of the justifications must be verified, but here goes:
ne^{-n\lambda}= \frac{d}{d\lambda}(-e^{-n\lambda}). The series e^{-n\lambda}is geometric, with ratio e^{-\lambda}, and thus \sum_{n=0}^{\infty}e^{-n\lambda}=\frac{1}{1-e^{-\lambda}}=\frac{e^{\lambda}}{e^{\lambda}-1}. Change the sign and derive with respect to λ and you are there.