How to sum $\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$? Does anyone know the general strategy for summing a series of the form:
$$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$
where $a$ is a positive integer?
Any hints or ideas would be great!

Use the fractional expansion of $\cot z$, you can get:
$$\frac{1}{e^t-1} -\frac{1}{t} +\frac{1}{2} =\sum_{n=1}^{\infty}\frac{2t}{t^2 +4n^2\pi^2} $$
