 Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3/7/f/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 Z} \frac{∣ u _{r} ∣ ^{2}}{V ^{2}} = \frac{8}{π ^{2}} \sum_{j = 0}^{\infty} \frac{1}{( 2 j + 1 ) ^{2}} .

Thus

j = 0 \sum \infty \frac{1}{( 2 j + 1 ) ^{2}} = \frac{π ^{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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