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

Find the value of $R$ for the power series

$$\sum_{n=1}^{\infty} \frac{z^{n}}{n}$$