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# Euler's number

Euler's number $\scriptstyle e$, sometimes called Napier's constant, is the base of the exponential function and the natural logarithm. $\scriptstyle e$ is transcendental.

The exponential function is the eigenfunction (with eigenvalue 1) of the differential operator, i.e.

D(ex) = ex.

## Value

The decimal expansion of e is

$e = 2.71828182845904523536028747135266249775724709\ldots$ (A001113).

Its continued fraction expansion is

$e = 2 ~+~ \cfrac{1}{1+\cfrac{1}{{\mathbf 2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1} {{\mathbf 4} +\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{{\mathbf 6}+\cfrac{1} {1+\cfrac{1}{1+\ddots}}}}}}}}}},$
$\quad \frac{1}{e - 1} = {\mathbf 0} ~+~ \cfrac{1}{1+\cfrac{1}{1+\cfrac{1} {{\mathbf 2}+\cfrac{1}{1+\cfrac{1}{1 +\cfrac{1}{{\mathbf 4}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{{\mathbf 6}+\cfrac{1}{1+\ddots}}}}}}}}}}$

or

$e = [2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \textbf{10}, 1, 1, \textbf{12}, 1, 1, \textbf{14}, 1, 1, \textbf{16}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots].$ (A003417).

## Formulas

$e = \sum_{n = 0}^{\infty} \frac{1}{n!}$
• $e = \lim_{x \to 0} (1+x)^{1/x} = \lim_{x \to \infty} (1+{1/x})^{x}.$
• $e = \lim_{n \to \infty} \sqrt[n]{\frac{n^n}{n!}} = \lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}}$

which results from Stirling's approximation.

• $e = \lim_{n \to \infty} \frac{n!}{!n}$

where $\scriptstyle n!$ is the factorial and $\scriptstyle !n$ is the subfactorial.

## 1/(e-1)

$\frac{1}{e-1} = \sum_{n=1}^{\infty} e^{-n}$

Its decimal expansion is

$\frac{1}{e - 1} = 0.581976706869326424385002005109011558546869301075 \ldots$ (A073333)
$\frac{1}{e - 1} = 0 + \cfrac{1}{1 + \cfrac{2}{2 + \cfrac{3}{3 + \cfrac{4}{\ddots}}}}$ (A110654)

## Power towers

$\scriptstyle n$ $\scriptstyle e$↑↑$\scriptstyle n$[1] Decimal expansion A-number
0 $\scriptstyle e$↑↑0 empty product 1
1 $\scriptstyle e$↑↑1 $\scriptstyle e$ 2.71828182846... A001113
2 $\scriptstyle e$↑↑2 $\scriptstyle e^e$ 15.15426224147926418976... A073226
3 $\scriptstyle e$↑↑3 $\scriptstyle e^{e^e}$ 3814279.1047602205922... A073227
4 $\scriptstyle e$↑↑4 $\scriptstyle e^{e^{e^e}}$ 2.331504399... $\times 10^{1656520}$ A085667

A004002 Benford numbers: a(n) = e^e^...^e (n times, n ≥ 0) rounded to nearest integer.

{1, 3, 15, 3814279, ...}

## Notes

1. See Knuth's arrow notation and tetration.