= Fourier series of a periodically extended quadratic arch
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{title2=$f_{\rm per}(x)=\frac16-\frac1{\pi^2}\sum_{n\ge1}\frac{\cos(2\pi nx)}{n^2}$}
The period-one extension of $x(1-x)$ on $[0,1]$ has <Fourier series> $1/6-\pi^{-2}\sum_{n\ge1}n^{-2}\cos(2\pi nx)$. Endpoint agreement makes the periodic function continuous. Its derivative series is $(2/\pi)\sum_{n\ge1}n^{-1}\sin(2\pi nx)$, which equals $1-2x$ in the open unit interval and converges to zero at integers, the midpoint of the derivative's jump. <Termwise differentiation of a Fourier series> applies locally away from the corners.
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