I was wondering if there is an interpretation or specific meaning to the series expansion of $e$.
$$ e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \cdots $$
Do the factorials in the denominator have something to do with the compound interest formula or the limit definition of $e$?
Additionally, I would really appreciate it if you provided me with something fun and curious that has to do with $e$.
Thanks!
The limit definition of $e,$ which arises in the study of compound interest, is
$$\lim_{n\to \infty} \left(1+\frac 1 n\right)^n.$$
The binomial series for $\left(1+\frac1n\right)^n$ is $$1 + n \frac 1 n + \frac {n(n-1)}{2!}\biggl(\frac 1n\biggr)^2 + \frac{n(n-1)(n-2)}{3!}\biggl(\frac 1 n\biggr)^3+...;$$
I think you could see from that where the factorials come from.
I think most mathematicians would agree with @Gerry Myerson's comment that the most beautiful
equation involving $e$ is Euler's identity:
$$e^{i \pi} + 1 = 0,$$ but here's another fun fact about Euler's number, involving the factorial function:
$$e=\lim_{n\to\infty}\frac n {\sqrt[n]{n!}}.$$