 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ia/paper-4/5f/d/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Every term is nonnegative. Since

1 - cos x \leq x^{2} /2

0 \leq 1 - cos (π / k) \leq \frac{π ^{2}}{2 k ^{2}} .

Comparison with

\sum k^{- 2}

shows that the partial sums are bounded. Being increasing, they converge to the finite limit

k = 1 \sum \infty (1 - cos (π / k)) .

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