Series of Power over Factorial Converges

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

Proof

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


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

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, $\displaystyle \sum_{n=0}^\infty \frac {x^n} {n!}$ 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 Exponential Definitions, where it is shown that this series converges to the exponential function.