I am having problems showing that $$\sum_{n=1}^\infty\frac{1}{1+n^2}=\frac{1}{2}(\pi\coth\pi-1)\quad $$ Here's my attept to this point: I tried to express each term using a partial fraction decomposition: $\frac1{1+ni}$ and $\frac1{1-ni}$. And I've been suggested from there that partial fractions can be expressed in terms of hyperbolic trigonometric functions. But i am stuck at this point. How to write a solution for this sum?
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1Do you have to perform the summation explicitly, using no other methods? Because this is the Mittag-Leffler series for $\coth$, and the standard way to arrive to this result is to use the Mittag-Leffler theorem and the calculus of residues. Here you can find my answer explaining the procedure. – Daigaku no Baku Jan 24 '24 at 22:55