Series expression for inverse hyperbolic function

[gdumont]

Hi,

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

 

[hypermorphism]

Do you know the sum of the geometric series ?

 

[gdumont]

I realized 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!