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 $$n \to \infty$$.

This follows from the results:
 * Power of Reciprocal, where $$\frac 1 n \to 0$$ as $$n \to \infty$$;
 * The Squeeze Theorem for Sequences, as $$\frac 1 {n + 1} < \frac 1 n$$;
 * The Combination Theorem for Sequences: Sum of Limits, putting $$\lambda = \left|{x}\right|$$ and $$\mu = 0$$.

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.

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