Define the radius of convergence $R$ of a complex power series $\sum a_{n} z^{n}$. Prove that $\sum a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

If $\left|a_{n}\right| \leqslant\left|b_{n}\right|$ for all $n$ does it follow that the radius of convergence of $\sum a_{n} z^{n}$ is at least that of $\sum b_{n} z^{n}$ ? Justify your answer.