= Solution
{c}
The <radius of convergence> of a <power series> $\sum_{n=0}^{\infty}a_nz^n$ is the number $R\in[0,\infty]$ such that the <series> has <absolute convergence> for $|z|<R$ and diverges for $|z|>R$. Equivalently,
$$
\boxed{R=\sup\left\{r\geq0:\ \sum_{n=0}^{\infty}|a_n|r^n<\infty\right\}.}
$$
Use $r^0=1$, including at $r=0$. The <supremum> may be $+\infty$. There is no general conclusion at $|z|=R$: boundary convergence depends on the <coefficients>. If $R=0$, only the centre is guaranteed; if $R=\infty$, every finite $z$ lies inside the <disk of convergence>.
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