Series of Power over Factorial Converges

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


Also see


Sources