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.

$\blacksquare$

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 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.

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 6.19$