Definition:Interval of Convergence

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

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

Then the set of values of $x$ for which $S \left({x}\right)$ converges is called the interval of convergence of $S$.