Ask about beautiful properties of $e$ One of students asked me about "some beautiful properties (or relation) of $e$". Then I list like below 
\begin{align}
& e \equiv \lim_{x \to \infty} \left(1+\frac{1}{x} \right)^x\\[10pt]
& e = \sum_{k=0}^\infty \frac{1}{k!}\\[10pt]
& \frac{d}{dx} (e^x) = e^x\\[10pt]
& e^{ix} = \cos x  + i \sin x  \quad \text{(Euler)}\\[10pt]
& e^{i \pi} + 1 = 0
\end{align} After this, he asked me for more relation or properties. I said 
I'll think and answer ...
Now I want help to add some relation, properties, or visual things (like proof without words)
Please help me to add something more. Thanks in advance.
***The class was math. 1. engineering

Here is the (simple) continued fraction for $e-1.$ The pattern continues forever, 1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...
To get $e$ itself add one, this is the same as changing the first $1$ to a $2$
$$  
\begin{array}{ccccccccccccccccccccccc}
 & & 1 & & 1 & & 2 & & 1 & & 1 & & 4 & & 1 & & 1 & & 6 &   \\
\frac{0}{1} & \frac{1}{0} & & \frac{1}{1} &  &  \frac{2}{1} & &  \frac{5}{3}  & & \frac{7}{4} & &  \frac{12}{7} & &   \frac{55}{32}  & &   \frac{67}{39}  & &   \frac{122}{71}   & &   \frac{799}{465}  \\
\end{array}
$$
