An Unexpected Number

Let us define $a_n = \dfrac {1+ \frac {1+\frac \cdots{n+2}}{n+1}}n$ and the value of the expression be $x$ . Then

$\begin{aligned} x & = 1 + a_1 \\ & = 1 + 1 + a_2 \\ & = 1 + 1 + \frac {1+a_3}2 \\ & = 1 + 1 + \frac 12 + \frac {\frac {1+a_4}3}2 \\ & = 1 + 1 + \frac 12 + \frac 1{3!} + \frac {\frac {1+a_5}4}{3!} \\ & = \frac 1{0!} + \frac 1{1!} + \frac 1{2!} + \frac 1{3!} + \frac 1{4!} + \cdots \\ & = e \approx \boxed{2.718} \end{aligned}$

The partial sum with top term $1+\frac{1}{N}$ comes out to be $\sum_{k=0}^{N} \frac{1}{k!}$ after a bit of simplification. In the example displayed, with $N=5$ , the sum is $\sum_{k=0}^{5} \frac{1}{k!}=\frac{163}{60}$ . The limit is $\boxed{e}$ .

Very nice puzzle, but I would claim that the appearance of Euler's number is never totally "unexpected." ;) One of my profs used to say that $e$ is the "super number" and $e^x$ is the "super function."

This is simply the Engel expansion of Euler's number.......!!!