\begin{definition}[Definition:Radius of Convergence/Real Domain]
Let $\xi \in \R$ be a real number.
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.
Let $I$ be the interval of convergence of $\map S x$.
Let the endpoints of $I$ be $\xi - R$ and $\xi + R$.
(This follows from the fact that $\xi$ is the midpoint of $I$.)
Then $R$ is called the '''radius of convergence''' of $\map S x$.
If $\map S x$ is convergent over the whole of $\R$, then $I = \R$ and thus the radius of convergence is infinite.
\end{definition}