 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/6/b/iii/solution

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

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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