Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 7 / d

The denominator vanishes at $z=\pm i\sqrt{2}$, giving two simple poles. Their distances from the center are both $|1\mp i\sqrt{2}|=\sqrt{3}$. There are no nearer singularities, so the Taylor series has radius of convergence

Apply the Cauchy-Hadamard theorem to the power series coefficients $a_n=[n/(n+\alpha ){]}^{n^2}$ for $n\ge 1$:For example, taking logarithms gives $-n\log (1+\alpha /n)\to -\alpha$. ThereforeAny convention for the isolated $n=0$ coefficient has no effect on the radius of convergence.