Last week I tried to find a proof for the so called partial fraction decomposition of the cotangent, that is
$$\pi \, \cot(\pi k) = \sum_{m=-\infty}^{\infty} \frac{k}{k^2-m^2}$$
For the proof I used Fourier-series in the following way:
An even function $f(w)=f(-w)$ fulfills
$$2 \pi f(z) = \sum_{m=-\infty}^\infty \cos(m z) \left( \int_{-\pi}^\pi f(w) \cos(m w) dw \right)$$
Now I take the even function $f(z) = \cos(k z)$ and get for the Fourier-coefficients:
$$a_m = \int_{-\pi}^{\pi} \cos(m w) \cos(k w) dw = \frac{2 \sin(\pi k) (-1)^m k}{k^2-m^2}$$
It follows, that
$$2 \pi \cos(k z) = \sum_{m=-\infty}^\infty \frac{2 \sin(\pi k) (-1)^m k \cos(m z)}{k^2-m^2}$$
Putting $z = \pi$ and using $\cos(m \pi) = (-1)^m$ gives the sought for formula.
This seems to be more than a special ad-hoc trick, but a method to compute the sum of a lot of series by fourier expansion of suitable "kernels" (like $\cos(k z)$ above):
To sum $\sum_{m=-\infty}^\infty g(z,m)$ find a kernel $f(t,z)$ and a fourier expansion (for simplicity in formulas I consider $f(t,z)$ even in $t$):
$$2 \pi f(t,z) = \sum_{m=-\infty}^\infty g(z,m) h(z) \cos(m t)$$
so that
$$\int_{-\pi}^\pi f(w,z) \cos(m w) dw = g(z,m) h(z)$$
holds. Then
$$\sum_{m=-\infty}^\infty g(z,m) \cos(m t_0) = 2 \pi f(t_0,z)/h(z) $$
Now my question is: To compile in the answers a list of applications of the principle above to different examples from "practice".