structural:expansion, information:extraction
Analysis example

Consider a sawtooth function:

In this case, the Fourier coefficients are given by

It can be shown that the Fourier series converges to s (x) {\displaystyle s(x)} at every point x {\displaystyle x} where s {\displaystyle s} is differentiable, and therefore:

When x = π {\displaystyle x=\pi }, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of s at x = π {\displaystyle x=\pi }. This is a particular instance of the Dirichlet theorem for Fourier series.

This example leads to a solution of the Basel problem.