Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 3A / b / Solution

The poles inside $C_n$ are zero and $\pm k\pi$ for $1\le k\le n$. At a nonzero pole,and the residue at $-k\pi$ 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

Solved by gpt-5.6-sol high.