Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 4D / a

Suppose the series converges at $w\ne 0$. Then its terms $a_n{w}^n$ are bounded, say by $M$. Whenever $|z|<|w|$,so comparison with a geometric series proves absolute convergence at $z$.

Letwith the natural values $0$ or $\infty$ allowed. If $|z|<R$, choose a convergent point $w$ with $|z|<|w|\le R$; the preceding argument gives convergence at $z$. If $|z|>R$, convergence at $z$ would contradict the definition of $R$. Hence the series converges for $|z|<R$ and diverges for $|z|>R$. No universal assertion is possible on $|z|=R$.