Definition:Radius of Convergence

Let $$\xi \in \mathbb{R}$$ be a real number.

Let $$S \left({x}\right) = \sum_{n=0}^\infty a_n \left({x - \xi}\right)^n$$ 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 $$\mathbb{R}$$, then $$I = \mathbb{R}$$ and thus the radius of convergence is infinite.