 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-3/3a/b/solution

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

The poles inside

C_{n}

are zero and

\pm kπ

for

1 \leq k \leq n

. At a nonzero pole,

Res_{kπ} \frac{1}{z ^{3} s i n z} = \frac{( - 1 ) ^{k}}{k ^{3} π ^{3}},

and the residue at

- kπ

is its negative. At zero, the Laurent series contains only even powers because the function is even, so its residue is zero. All residues cancel and therefore

I_{n} = 0 (n = 0, 1, 2, \dots) .

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