 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-09-25

Taking real parts of the geometric sum gives

a_{n} = ℜ \frac{1 - e ^{i (n + 1) y}}{1 - e ^{i y}},

(a_{n})

is bounded when

cos y \neq = 1

. It cannot converge: convergence would imply

cos (n y) = a_{n} - a_{n - 1} \to 0

, but then

cos (2 n y) = 2 cos^{2} (n y) - 1 \to - 1,

whereas the same necessary condition applied to the subsequence

2 n

would give a limit of zero.

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