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$$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)} $$ $$\frac{1}{\sinh z}=\frac{1}{z}+\sum_{m\geq 1}\left(\frac{1}{z-m\pi i}+\frac{1}{z+m\pi i}\right)(-1)^m $$ $$\frac{1}{\sinh(\pi n)}=\frac{1}{\pi n}+\frac{1}{\pi}\sum_{m\geq 1}\left(\frac{1}{n-mi}+\frac{1}{n+mi}\right)(-1)^m $$ $$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\frac{7\pi^3}{720}+\frac{2}{\pi}\sum_{m\geq 1}\sum_{n\geq 1}\frac{(-1)^{n+m+1}}{n^2(n^2+m^2)}$$ $$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\frac{7\pi^3}{720}+\frac{2}{\pi}\sum_{m\geq 1}(-1)^{m+1}\left(\frac{1}{2m^4}-\frac{\pi^2}{12m^2}-\frac{\pi}{2m^3\sinh(m\pi)}\right)$$ $$2\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\frac{7\pi^3}{720}+\frac{2}{\pi}\sum_{m\geq 1}(-1)^{m+1}\left(\frac{1}{2m^4}-\frac{\pi^2}{12m^2}\right)$$ $$2\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\frac{7\pi^3}{720}-\frac{\pi^3}{240}$$ $$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\color{red}{\frac{\pi^3}{360}}.$$
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Tue, 09 Jan 2018 16:04 GMT