Radius of Convergence of Power Series over Factorial

Theorem
Let $$\xi \in \mathbb{R}$$ be a real number.

Let $$f \left({x}\right) = \sum_{n=0}^\infty \frac {\left({x - \xi}\right)^n} {n!}$$.

Then $$f \left({x}\right)$$ converges for all $$x \in \mathbb{R}$$.

That is, the interval of convergence of the power series $$\sum_{n=0}^\infty \frac {\left({x - \xi}\right)^n} {n!}$$ is $$\mathbb{R}$$.

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

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

$$ $$ $$ $$

Hence the result.