Summation Contest Conceptual Problem On Digamma Function

Since the function $\displaystyle f(z)=\frac{1}{z^2+4}$ which we integrate in the upper half plane.

$f(z)$ has simple poles at $z= \pm 2i$ but $2i$ only lies in the upper half and so we have the residues of $f(z)\cot(\pi z)$ at the poles of $f(z)$ .

$\displaystyle \text{Res(2i)} = \lim_{z\to 2i} \frac{\pi \cot(\pi z)}{z^2+4} =- \frac{\pi \coth(2\pi)}{4}$

Since the function is even we conclude by summation lemma ,

$\displaystyle \sum_{n=-\infty}^{\infty} \frac{1}{n^2+4} = -(-\frac{\pi \coth(2\pi)}{4})$ Since we have , $\displaystyle \sum_{n=-\infty}^{\infty} \frac{1}{n^2+4} = 2\sum_{n=0}^{\infty} \frac{1}{n^2+4}$ we deduce that ,

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2+4} = \frac{1}{8}(2\pi \coth(2\pi) -1)$