# Series of Power over Factorial Converges

 It has been suggested that this page or section be merged into Radius of Convergence of Power Series over Factorial. (Discuss)

## Theorem

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

## Proof

If $x = 0$ the result is trivially true as:

$\forall n \ge 1: \dfrac {0^n} {n!} = 0$

If $x \ne 0$ we have:

$\left|{\dfrac{\left({\dfrac {x^{n+1}} {(n+1)!}}\right)}{\left({\dfrac {x^n}{n!}}\right)}}\right| = \dfrac {\left|{x}\right|} {n+1} \to 0$

as $n \to \infty$.

This follows from the results:

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

$\blacksquare$

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

Another alternative is to view this as an example of Radius of Convergence of Power Series over Factorial setting $\xi = 0$.