Series of Power over Factorial Converges

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:
 * Power of Reciprocal, where $\dfrac 1 n \to 0$ as $n \to \infty$
 * The Squeeze Theorem for Sequences, as $\dfrac 1 {n + 1} < \dfrac 1 n$
 * The Combination Theorem for Sequences: Multiple Rule, putting $\lambda = \left|{x}\right|$.

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

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 Power Series over Factorial setting $\xi = 0$.

Also see

 * Equivalence of Definitions of Exponential Function, where it is shown that this series converges to the exponential function.