<s>[INST] 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
 [/INST] Here  are some more   relations which might be pleasing.

From section 1.3 of Mathematical Constants by S.R. Finch:
  
  
*
  
*A    Wallis-like  infinite   product is
  \begin{align*}
e=\frac{2}{1}\cdot\left(\frac{4}{3}\right)^{\frac{1}{2}}
\cdot\left(\frac{6\cdot 8}{5\cdot 7}\right)^{\frac{1}{4}}
\cdot\left(\frac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}\right)^{\frac{1}{8}}\cdots
\end{align*}
  
*From Stirling's formula we derive
  \begin{align*}
e=\lim_{n\rightarrow  \infty}\frac{n}{(n!)^{\frac{1}{n}}}
\end{align*}
  
*Another continued fraction representation is
  \begin{align*}
e&=2+\frac{\left.2\right|}{\left|2\right.}
+\frac{\left.3\right|}{\left|3\right.}
+\frac{\left.4\right|}{\left|4\right.}
+\frac{\left.5\right|}{\left|5\right.}+\cdots\\
&=2+\frac{2}{2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\cdots}}}}
\end{align*}

$$ $$

In section Intriguing Results in Real Infinite Series by D.D. Bonar and M.J. Khoury we  find
  
  
*
  
*Gem 89 (American Math Monthly 42:2 pp. 111-112)
  
  
  \begin{align*}
e=\frac{1}{5}\left(\frac{1^2}{0!}+\frac{2^2}{1!}+\frac{3^2}{2!}+\frac{4^2}{3!}+\cdots\right)
\end{align*}

</s>