Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a complex power series. Prove that there exists an $R \in[0, \infty]$ such that the series converges for every $z$ with $|z|<R$ and diverges for every $z$ with $|z|>R$.

Find the value of $R$ for each of the following power series: (i) $\sum_{n=1}^{\infty} \frac{1}{n^{2}} z^{n}$; (ii) $\sum_{n=0}^{\infty} z^{n !}$.

In each case, determine at which points on the circle $|z|=R$ the series converges.