Past exam of the short options course of the University of Oxford / 2017 / 6 / b / iii

Table of contents
- Solution iii

Solution

The sine functions obey

\int_{0}^{π} sin (n x) sin (m x) d x = (π /2) δ_{nm}

. The Parseval identity for the Fourier sine series gives

1 = \int_{0}^{π} y (x)^{2} d x = \frac{π}{2} \sum_{n \geq 1} a_{n}^{2} .

Consequently

n \geq 1 \sum a_{n}^{2} = \frac{2}{π} .

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