= Solution
The <radius of convergence> of a <power series> is the extended real number $R\in[0,\infty]$ characterized by \b[absolute convergence for $|x|<R$ and divergence for $|x|>R$]. Equivalently,
$$
\boxed{R=\sup\{r\geq0:\ \text{the series converges absolutely for every real }|x|<r\}.}
$$
For $R=0$, only the central point is guaranteed; for $R=\infty$, every real point is inside. At finite endpoints $x=\pm R$, convergence must be investigated separately and is not determined by $R$.
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