Paper 1, Section I, D

Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $R \in[0, \infty]$.

Suppose the power series has radius of convergence $R$, with $0<R<\infty$. Show that the sequence $\left|a_{n} z^{n}\right|$ is unbounded if $|z|>R$.

Find the radius of convergence of $\sum_{n \geqslant 1} z^{n} / n^{3}$.

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