# Radius of Convergence of Power Series over Factorial

## Theorem

### Real Case

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

Let $\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!}$.

Then $\map f x$ converges for all $x \in \R$.

That is, the interval of convergence of the power series $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!}$ is $\R$.

### Complex Case

Let $\xi \in \C$ be a complex number.

Let $\ds \map f z = \sum_{n \mathop = 0}^\infty \dfrac {\paren {z - \xi}^n} {n!}$.

Then $\map f z$ converges absolutely for all $z \in \C$.

That is, the radius of convergence of the power series $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {z - \xi}^n} {n!}$ is infinite.