The power series for $e^x$ is as follows.
$$e^{x} =\sum ^{\infty }_{n=0}\frac{x^{n}}{n!}$$
If we define $n! = \Gamma(n+1)$, then we have
$$n!=\int ^{\infty }_{0} x^{n} e^{-x} dx.$$
An extremely surface level observation is that if we compare the summand/integrand of each equation to their corresponding left-hand-sides, we end up with the following correspondences.
$$e^x \longleftrightarrow \frac{x^n}{n!} \hspace{2 in} n! \longleftrightarrow \frac{x^n}{e^x}$$
Very naïvely, one could make the observation that the correspondence on the left looks like an algebraic manipulation of the correspondence on the right (multiply both sides by $n!$ and divide by $e^x$).
Is this merely coincidence or is there some deeper connection here?