I faced a problem with one very interesting series: $\sum_{n=1}^\infty \frac{1}{n^3\sin{\pi\sqrt{2}n}}$ I know that the answer is $-\frac{13\pi^3}{360\sqrt{2}}$, but I couldn’t get it.
At first I have tried the fraction decomposition of the function $\frac{1}{\sin{\pi\sqrt{2}n}}$, but when I’ve done this I faced a new series that I can’t handle $\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{(-1)^k}{n^2(2n^2-k^2)}$
Next try was to just apply the residue theorem to first or second series, but I didn’t get anything good.
At last I’ve tried to find some function whose residues will be members of the series(just like you can do with this series $\sum_{n=1}^\infty \frac{1}{(\sinh{\pi n})^2}$)
So, any ideas or approaches will be appreciated! Especially some tricky methods, because I believe that this value can be obtained without “deep math”.
