Let
$$
r_0 = \sup \{\; r \ge 0 : \mbox{ there exists } M \ge 0 \mbox{ such that } |a_n|r^n \le M \mbox{ for all } n \ge 0 \}
$$
I'll leave the special cases where $r_0=0$ or $r_0=\infty$ to you. I'll assume $0 < r_0 < \infty$.
If $|z| < r_0$, then $\sum_{n}a_n z^n$ converges absolutely by the definition of $r_0$. Hence, the series converges absolutely for any fixed $z$ for which $|z| < r_0$.
If $|z| > r_0$, then $\{ |a_n||z|^n \}$ is an unbounded sequence, which means that the general term of the series $\sum_{n}a_n z^n$ does not converge to $0$ and, thus, cannot converge conditionally or absolutely.
By the definition of the radius of convergence, $r_0$ is the radius of convergence.