Series expression for inverse hyperbolic function

1. Nov 13, 2005

gdumont

Hi,

I'm trying to find a way to prove that
$$\sum_{n=1}^{\infty} n e^{-n x} = \frac{1}{4}\sinh^{-2} \frac{x}{2}$$
Any help greatly appreciated

2. Nov 13, 2005

hypermorphism

Do you know the sum of the geometric series ?

3. Nov 13, 2005

gdumont

I realised that it was the $\sinh x$ function and not the inverse function. And yes I used the geometric series to show the relation. Thanks anyways!