Show that if the power series $\sum_{n=0}^{\infty} a_{n} z^{n}(z \in \mathbb{C})$ converges for some fixed $z=z_{0}$, then it converges absolutely for every $z$ satisfying $|z|<\left|z_{0}\right|$.

Define the radius of convergence of a power series.

Give an example of $v \in \mathbb{C}$ and an example of $w \in \mathbb{C}$ such that $|v|=|w|=1, \sum_{n=1}^{\infty} \frac{v^{n}}{n}$ converges and $\sum_{n=1}^{\infty} \frac{w^{n}}{n}$ diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{z^{n}}{n}$.