Sawtooth wave

Example

Let $f$ be the $2\pi$-periodic function with $f(x) = \frac{\pi - x}{2}$ for $0 < x < 2\pi$ and $f(0) = 0$. Its graph is a sawtooth: it falls linearly from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$ on each period and jumps back up at the multiples of $2\pi$. It is odd, so its cosine coefficients vanish, and integration by parts gives for $n \ge 1$ $$b_n = \frac{1}{\pi} \int_0^{2\pi} \frac{\pi - x}{2} \sin nx \, dx = \frac{1}{\pi} \cdot \frac{1}{2} \cdot \frac{2\pi}{n} = \frac{1}{n}.$$ So its Fourier series is $$\sum_{n=1}^{\infty} \frac{\sin nx}{n}.$$ The function has bounded variation, so by the Dirichlet conditions the series converges to $\frac{\pi - x}{2}$ for $0 < x < 2\pi$, and to $0$ at the jumps $x \in 2\pi\Z$.