Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 13A / b / Solution

Using the Taylor series of the exponential and series division,Thus the first three nonzero Laurent terms areThe singularities occur at $z=k\pi i$. The nearest nonzero ones are $\pm \pi i$, so this Laurent series is valid on $0<|z|<\pi$.