Reciprocal-square sine series positivity ($S(\xi )={\sum }_{l\ge 1}\sin (l\xi )/l^2$)

The absolutely convergent sine series $S(\xi )={\sum }_{l\ge 1}\sin (l\xi )/l^2$ 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.