 Fourier series of a half-wave rectified ramp

Extend $f(x)=\max (0,2x)$ on $(-1,1)$ with period two. Its Fourier coefficients in the basis $\cos (n\pi x),\sin (n\pi x)$ are $a_0=1$, $a_n=2[(-1{)}^n-1]/(n\pi {)}^2$, and $b_n=2(-1{)}^{n+1}/(n\pi )$, by integration by parts on $(0,1)$. ThusIts even part is the periodic extension of $|x|$ and its odd part is the Fourier series of a periodic sawtooth for $x$. The cosine terms converge absolutely and uniformly; the sine terms account for the jump at odd integers. At each jump, the full Fourier series has value $1$, the average of the two one-sided limits $0$ and $2$.