Definition:Radius of Convergence/Real Domain

Definition
Let $\xi \in \R$ be a real number.

Let $\displaystyle S \left({x}\right) = \sum_{n \mathop = 0}^\infty a_n \left({x - \xi}\right)^n$ be a power series about $\xi$.

Let $I$ be the interval of convergence of $S \left({x}\right)$.

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 $S \left({x}\right)$.

If $S \left({x}\right)$ is convergent over the whole of $\R$, then $I = \R$ and thus the radius of convergence is infinite.