Factorial is not of Exponential Order

Theorem

Let $\Gamma$ denote the gamma function.

Let $\map f t = \map \Gamma {t + 1} = t!$.

Then:

$f$ is not of exponential order.

That is, it grows faster than any exponential.

Proof

From Gamma Function is Continuous on Positive Reals, $f$ is continuous for $t \ge 0$.

Set $t > 0$.

From Stirling's Formula:

$t! \sim \sqrt {2 \pi t} \paren {\dfrac t e}^t$

where $\sim$ denotes asymptotic equality.

That is,

\(\ds t!\)

\(\sim\)

\(\ds \sqrt {2 \pi t} \paren {\frac t e}^t\)

\(\ds \)

\(>\)

\(\ds t^t \, e^{-t}\)

Aiming for a contradiction, suppose $t^t \, e^{-t}$ is of exponential order $a$.

Then, for $t$ sufficiently large, there exists a $K > 0$ such that:

\(\ds t^t \, e^{-t}\)

\(<\)

\(\ds K e^{a t}\)

\(\ds \leadsto \ \ \)

\(\ds t^t\)

\(<\)

\(\ds K e^{\paren {a + 1} t}\)

This implies that $t^t$ is of exponential order, which is false.

From this contradiction it follows that $t!$ is also not of exponential order.

$\blacksquare$