 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ia/paper-1/3f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Separate off the integer part of the phase. Set

d_{n} = n^{2} + n - n

. Rationalization gives

d_{n} = \frac{1}{1 + 1/ n + 1} ⟶ \frac{1}{2} .

Since squaring removes the sign in

cos (πn + u) = (- 1)^{n} cos u

, the given sequence satisfies

b_{n} = cos^{2} (π d_{n})

. The continuity of the cosine now gives

n \to \infty lim b_{n} = cos^{2} (π /2) = 0.

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