Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $0 \leqslant R \leqslant \infty$.

Find the radius of convergence of $\sum_{n \geqslant 0} p(n) z^{n}$, where $p(n)$ is a fixed polynomial in $n$ with coefficients in $\mathbb{C}$.