Editing Fourier series (section)

==== Example ====
[[File:sawtooth pi.svg|thumb|right|400px|Plot of the [[sawtooth wave]], a periodic continuation of the linear function <math>s(x)=x/\pi</math> on the interval <math>(-\pi,\pi]</math>]]

[[File:Periodic identity function.gif|thumb|right|400px|Animated plot of the first five successive partial Fourier series]]

Consider a sawtooth function:
<math display="block">s(x) = s(x + 2\pi k)  = \frac{x}{\pi}, \quad \mathrm{for } -\pi < x < \pi,\text{ and } k \in \mathbb{Z}.</math>
In this case, the Fourier coefficients are given by
<math display="block">\begin{align}
a_0 &= 0.\\
a_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \cos(nx)\,dx = 0, \quad n \ge 1. \\
b_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \sin(nx)\, dx\\
&= -\frac{2}{\pi n}\cos(n\pi) + \frac{2}{\pi^2 n^2}\sin(n\pi)\\
&= \frac{2\,(-1)^{n+1}}{\pi n}, \quad n \ge 1.\end{align}</math>
It can be shown that the Fourier series converges to <math>s(x)</math> at every point <math>x</math> where <math>s</math> is differentiable, and therefore:
<math display="block">\begin{align}
s(x) &= a_0 + \sum_{n=1}^\infty \left[a_n\cos\left(nx\right)+b_n sin\left(nx\right)\right] \\[4pt]
&=\frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for}\ (x-\pi)\ \text{is not a multiple of}\ 2\pi.
\end{align}</math>
When <math>x=\pi</math>, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of <math>s</math> at <math>x=\pi</math>. This is a particular instance of the [[Convergence of Fourier series#Convergence at a given point|Dirichlet theorem]] for Fourier series.

This example leads to a solution of the [[Basel problem]].