Reciprocal-square sine series positivity
= Reciprocal-square sine series positivity
{title2=$S(\xi)=\sum_{l\ge1}\sin(l\xi)/l^2$}
The absolutely convergent sine series $S(\xi)=\sum_{l\ge1}\sin(l\xi)/l^2$ has the integral representation $S(\xi)=\sin\xi\int_0^\infty t e^{-t}/(1-2e^{-t}\cos\xi+e^{-2t})\,dt$. The integrand is positive for $0<\xi<\pi$, proving $S>0$ there without incorrectly maximizing individual Fourier terms.