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I am trying to prove $$\sum_{n\geq1}\frac1{n^2+1}=\frac{\pi\coth\pi-1}2$$ Letting $$S=\sum_{n\geq1}\frac1{n^2+1}$$ we recall the Fourier series for the exponential function $$e^x=\frac{\sinh\pi}\pi+\frac{2\sinh\pi}\pi\sum_{n\geq1}\frac{(-1)^n}{n^2+1}(\cos nx-n\sin nx)$$ Plugging in $x=\pi$ $$e^\pi=\frac{\sinh\pi}\pi+\frac{2\sinh\pi}\pi\sum_{n\geq1}\frac{(-1)^n}{n^2+1}(\cos n\pi-n\sin n\pi)$$ $$e^\pi=\frac{\sinh\pi}\pi+\frac{2\sinh\pi}\pi\sum_{n\geq1}\frac{(-1)^n}{n^2+1}((-1)^n-n\cdot0)$$ $$e^\pi=\frac{\sinh\pi}\pi+\frac{2\sinh\pi}\pi S$$ $$S=\frac{\pi e^\pi}{2\sinh\pi}-\frac12$$ But that is nowhere near to correct. What did I do wrong, and how do can I prove the identity? Thanks.

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