Radius of Convergence of Power Series over Factorial/Real Case

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

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

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

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

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

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

Hence the result.