 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/12/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

x = 0

, where

f (0) = 1

. Every cosine is one, so

1 = \frac{1}{2} + \frac{4}{π ^{2}} \sum_{m = 0}^{\infty} \frac{1}{( 2 m + 1 ) ^{2}}, m = 0 \sum \infty \frac{1}{( 2 m + 1 ) ^{2}} = \frac{π ^{2}}{8} .

The convergence established above justifies substituting this point into the series.

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