 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-2/17/c/solution

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

The periodic ramp is an odd function, so

a_{0} = a_{n} = 0

b_{n} = \frac{2}{π} \int_{0}^{π} x sin (n x) d x = \frac{2 ( - 1 ) ^{n + 1}}{n} .

x = 2 n = 1 \sum \infty \frac{( - 1 ) ^{n + 1}}{n} sin (n x), - π < x < π .

The periodic function is smooth at

x = π /2

, so the Dirichlet-Jordan convergence theorem permits evaluation there. Even-index terms vanish and odd-index sines alternate in sign, giving

1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \frac{π}{4} .

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