I am looking to show: $\sum_{n=1}^∞ \frac{\coth(nπ)}{n^3} = \frac{7π^3}{180}$
There is a hint earlier that you are supposed to be using the function $f(z)=\frac{\cot z\coth z}{z^3}$. I have calculated the residue at the pole of order 5 at $z=0$ as $-\frac{7}{45}$, but I am unsure how to calculate the other residues, so I can use the residue theorem.
I think there is a simple pole whenever $z=\frac{(2n+1)π}{2}$, as this is when $\cot z=0$ but I just don't know how to find the residue here. I'm presuming my residues will lead to the sum I'm wanting to find coming out in some form when I apply the residue theorem, but I'm just not sure how to get there.
Thanks so much for any help in advance.