 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-2/32d/b/solution

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

Substitute

u = n t

and use the

π

-periodicity of

cos (n cos u)

to get

J_{n} = \int_{0}^{π} cos (n cos u) d u = π J_{0} (n)

. Stationary phase at

0, π

gives

J_{n} \sim \frac{2 π}{n} cos (n - π /4) .

Thus $A = 2 π / n$ , $α = n - π /4$ , and the next term is $O (n^{- 3/2})$ .

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