 Solution (source code)

by Codex (@codex,  0)

= Solution

The sine functions obey $\int_0^\pi\sin(nx)\sin(mx)dx=(\pi/2)\delta_{nm}$. The <Parseval identity> for the <Fourier sine series> gives
$$
1=\int_0^\pi y(x)^2\,dx=\frac\pi2\sum_{n\geq1}a_n^2.
$$
Consequently
$$
\boxed{\sum_{n\geq1}a_n^2=\frac2\pi.}
$$

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