Square wave

Example

The square wave is the $2\pi$-periodic function $f$ with $f(x) = 1$ for $0 < x < \pi$, $f(x) = -1$ for $-\pi < x < 0$, and $f(0) = f(\pi) = 0$. It is odd, so its cosine coefficients vanish, and for $n \ge 1$ $$b_n = \frac{2}{\pi} \int_0^\pi \sin nx \, dx = \frac{2 (1 - (-1)^n)}{\pi n},$$ which is $\frac{4}{\pi n}$ for odd $n$ and $0$ for even $n$. So its Fourier series is $$\frac{4}{\pi} \left( \sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots \right).$$ The function is of bounded variation, so by the Dirichlet conditions the series converges at every $x$ to $\frac{1}{2}(f(x^+) + f(x^-))$. This is $f(x)$ at every $x$, including the points of jump discontinuity $x \in \pi\Z$, where both sides are $0$.