Power over Factorial

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Theorem

Let $x \in \R: x > 0$ be a positive real number.

Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as $x_n = \dfrac {x^n} {n!}$.

Then $\left \langle {x_n} \right \rangle$ converges to zero.

We need to show that $x_n \to 0$ as $n \to \infty$.

Let $N \in \N$ be the smallest natural number which satisfies $N > x$.

(From the Archimedean Principle, such an $N$ always exists.)

First we show that:

$\displaystyle \forall n > N: \frac {x^n} {n!} \le \frac {x^{N-1}} {\left({N-1}\right)!} \left({\frac {x} {N}}\right)^{n - N + 1}$

Note that as $N > x$, $\dfrac x N < 1$.

Also, $\displaystyle m > n \implies \frac x m < \frac x n$.

Thus:

$\displaystyle $	$\displaystyle \frac {x^n} {n!}$	$=$	$\displaystyle \frac x 1 \frac x 2 \cdots \frac x {N-1} \frac x N \frac x {N+1} \cdots \frac x n$	$\displaystyle $
$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {x^{N-1} } {\left({N-1}\right)!} \frac x N \frac x {N+1} \cdots \frac x n$	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\le$	$\displaystyle \frac {x^{N-1} } {\left({N-1}\right)!} \left({\frac x N}\right)^{n - N + 1}$	$\displaystyle $	as $\displaystyle \frac x {N+1}, \frac x {N+2}, \ldots, \frac x n < \frac x N$
$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {x^{N-1} } {\left({N-1}\right)!} \left({\frac x N}\right)^{1 - N} \left({\frac x N}\right)^n$	$\displaystyle $

As $\dfrac x N < 1$, it follows from Power of a Number Less Than One that $\left({\dfrac x N}\right)^n \to 0$ as $n \to \infty$.

For a given $x$ and $N$, $\displaystyle \frac {x^{N-1}} {\left({N-1}\right)!} \left({\frac x N}\right)^{1 - N}$ is constant.

$\displaystyle \frac {x^{N-1}} {\left({N-1}\right)!} \left({\frac x N}\right)^{1 - N} \left({\frac x N}\right)^n \to 0$ as $n \to \infty$

As (from above):

$\displaystyle \frac {x^n} {n!} \le \frac {x^{N-1}} {\left({N-1}\right)!} \left({\frac x N}\right)^{1 - N} \left({\frac x N}\right)^n$

the result follows from the Squeeze Theorem.

\(\displaystyle \)	\(\displaystyle \frac {x^n} {n!}\)	\(=\)	\(\displaystyle \frac x 1 \frac x 2 \cdots \frac x {N-1} \frac x N \frac x {N+1} \cdots \frac x n\)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {x^{N-1} } {\left({N-1}\right)!} \frac x N \frac x {N+1} \cdots \frac x n\)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle \frac {x^{N-1} } {\left({N-1}\right)!} \left({\frac x N}\right)^{n - N + 1}\)	\(\displaystyle \)	as $\displaystyle \frac x {N+1}, \frac x {N+2}, \ldots, \frac x n < \frac x N$
\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {x^{N-1} } {\left({N-1}\right)!} \left({\frac x N}\right)^{1 - N} \left({\frac x N}\right)^n\)	\(\displaystyle \)