Euler's number

From OeisWiki

Current revision (unreviewed)

This article needs more work.

Please help by expanding it!

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 (e x) = e x .

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)

with generalized continued fraction

$\frac{1}{e - 1} = 0 + \cfrac{1}{1 + \cfrac{2}{2 + \cfrac{3}{3 + \cfrac{4}{\ddots}}}}$ (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