Is there a closed-form equation for $n!$? If not, why not?
I know that the Fibonacci sequence can be described via the Binet's formula.
However, I was wondering if there was a similar formula for $n!$.
Is this possible? If not, why not?
If you're willing to accept an integral as an answer, then $n! = \int_0^\infty t^n e^{-t} \: dt$.
The relative error of Stirling's approximation gets arbitrarily small as n gets larger.
$$n!\sim\sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$
However, it is only an approximation, not a closed-form of $n!$