Series of Power over Factorial Converges

Theorem
The series $$\sum_{n=0}^\infty \frac {x^n} {n!}$$ converges for all real values of $$x$$.

Proof

 * If $$x = 0$$ the result is trivially true as $$\forall n \ge 1: \frac {x^n} {n!} = 0$$.


 * If $$x \ne 0$$ we have $$\left|\frac{\left({\frac {x^{n+1}} {(n+1)!}}\right)}{\left({\frac {x^n}{n!}}\right)}\right| = \frac {\left|{x}\right|} {n+1} \to 0$$ as $$n \to \infty$$.

Hence by the Ratio Test, $$\sum_{n=0}^\infty \frac {x^n} {n!}$$ converges.

Alternatively, the Comparison Test could be used but this is more cumbersome in this instance.