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!

Hint: Differentiate the natural logarithm of Euler's infinite product formula for the sine function, then use the relation between trigonometric and hyperbolic functions.
