Radius of Convergence of Power Series over Factorial/Complex Case

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

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

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

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

Proof
This is a power series in the form $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ where $\sequence {a_n} = \sequence {\dfrac 1 {n!} }$.

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

Hence the result.