Definition:Interval of Convergence

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

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