Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of complex numbers. Prove that there exists $R \in[0, \infty]$ such that the power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

Give an example of a power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ that diverges if $z=\pm 1$ and converges if $z=\pm \mathrm{i}$.