= Fourier series of a periodic sawtooth
{c}
The period-$L$ extension of $f(x)=Kx/L$ on $-L/2<x<L/2$ has <Fourier coefficients> $a_0=a_n=0$ and $b_n=K(-1)^{n+1}/(\pi n)$. Consequently
$$
f(x)\sim\frac K\pi\sum_{n=1}^\infty\frac{(-1)^{n+1}}n\sin\frac{2\pi nx}{L}.
$$
The <Fourier series> converges to the ramp away from its jumps and to zero at every jump, the average of the one-sided limits. For $K\ne0$, <Termwise differentiation of a Fourier series> fails here as an ordinary convergent series: the differentiated summands have nondecaying amplitude $2K/L$. The classical derivative equals $K/L$ between jumps, whereas the periodic <distributional derivative> includes negative point masses of weight $K$ at the jumps.
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