How to calculate $\sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$

Question

How can I calculate the summation: $\displaystyle \sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$

Answer 1

Hint: Apply the Residue theorem for $$f(z)=\frac{\pi\cot \pi z}{1+z^2}$$ using the circle at $0$ of radius $n+\frac12$ and take the limit.

Answer 2

Using Fourier Series, you can show that:

$$ \cot(\pi z) = \frac 1\pi \left( \frac 1z - \sum_{k=1}^\infty \frac{2z}{k^2 - z^2} \right) $$

Let $z = i$ to essentially get your sum, modulo some small simplifications which I'll leave to you.