Decreasing function that aproaches $e$

Question

I need an example of a decreasing function $f(x)$ for which it holds: $$ \lim_{x\to +\infty} f(x) = e.$$

Edit: As an answer to the first comment, function $f(x)$ can't use $e$. It should have some nice form like its increasing counterparts: $$\sum_{i=1}^{x}\frac{1}{i!}$$ or: $$(1 + 1/x)^x.$$ Comment: By a nice form, I mean using basic operations as in the above expressions.

Answer 1

An interesting example is

$$ f(x) = \frac{(x+1)^{x+1}}{x^x} - \frac{x^x}{(x-1)^{x-1}} $$

It was proved that $\lim_{x\to\infty}f(x) = e$ by Harlan J. Brothers and John Knox (pdf). Not only is the function decreasing, it also converges much quicker than other functions like $f(x) = \sum^x_{k=1}\frac{1}{k!}$ or even $f(x) = \frac{x}{\sqrt[x]{x!}}$.

Answer 2

The function $f(x) = (1 - 1/x)^{-x}$ converges to $e$ and is monotonically decreasing.

You are free to add to it any decreasing function that goes to zero, such as $e^{-x}$.