Radius of Convergence of Power Series over Factorial/Complex Case

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

Let $\displaystyle f \left({z}\right) = \sum_{n \mathop = 0}^\infty \dfrac {\left({z - \xi}\right)^n} {n!}$.

Then $f \left({z}\right)$ converges for all $z \in \C$.

That is, the radius of convergence of the power series $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({z - \xi}\right)^n} {n!}$ is infinite.

Proof
This is a power series in the form $\displaystyle \sum_{n \mathop = 0}^\infty a_n \left({z - \xi}\right)^n$ where $\left \langle {a_n} \right \rangle = \left \langle {\dfrac 1 {n!}} \right \rangle$.

Applying Radius of Convergence from Limit of Sequence/Complex Case, we find that:

Hence the result.