= Solution
Use <Parseval identity for Fourier series> with the coefficient normalization above. Only <odd> nonzero indices contribute; summing the positive and negative indices gives
$$
1=\sum_{r\in\mathbb Z}\frac{|u_r|^2}{\mathrm{V^2}}
=\frac8{\pi^2}\sum_{j=0}^{\infty}\frac1{(2j+1)^2}.
$$
Thus
$$
\boxed{\sum_{j=0}^{\infty}\frac1{(2j+1)^2}=\frac{\pi^2}{8}.}
$$
The factor of two from the negative <frequencies> is essential. The discontinuities do not obstruct <Parseval's identity>, which is a statement about <L2 space>.
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