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# Euler's number

(Redirected from E)

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

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

 D (e  x ) = e  x.

## e

The decimal expansion of e is (A001113)

e = 2.71828182845904523536028747135266249775724709...

## e  −  1

The continued fraction expansion of e  −  1 is (A003417)

e − 1 = 1 +
1
1 +
1
2 +
1
1 +
1
1 +
1
4 +
1
1 +
1
1 +
1
6 +
1
1 +
1
1 +
 1 ⋱
,

or, using the compact notation for simple continued fractions,

 e − 1 = [1; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, ..., 1, 1, 2n, ...].

## Formulas

e =
 ∞ ∑ n  = 0
 1 n!
,
${\displaystyle {\begin{array}{l}\displaystyle {e=\lim _{x\to 0}(1+x)^{1/x}=\lim _{x\to \infty }\left(1+{\frac {1}{x}}\right)^{x},}\end{array}}}$
${\displaystyle {\begin{array}{l}\displaystyle {e=\lim _{n\to \infty }{\left({\frac {n^{n}}{n!}}\right)}^{1/n}=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}},}\end{array}}}$

which results from Stirling’s approximation

e =
 n! !n
,

where n! is the factorial and !n is the subfactorial.

## 1/e

The decimal expansion of 1/e is (A068985)

 1 e
= 0.3678794411714423215955237701614608674458111310317678...

## 1/(e  −  1)

 1 e − 1
=
 ∞ ∑ n  = 1
e  − n.

Its decimal expansion is (A073333)

 1 e − 1
= 0.581976706869326424385002005109011558546869301075...

Its simple continued fraction expansion is

 1 e − 1
= 0 +
1
1 +
1
1 +
1
2 +
1
1 +
1
1 +
1
4 +
1
1 +
1
1 +
1
6 +
1
1 +
1
1 +
 1 ⋱
.

One of its generalized continued fraction expansions is (A110654)

 1 e − 1
=
1
1 +
2
2 +
3
3 +
 4 ⋱

## Power towers

n e ↑↑ n[1] Decimal expansion A-number
0 e ↑↑ 0 empty product 1
1 e ↑↑ 1 e 2.71828182846... A001113
2 e ↑↑ 2 ee 15.15426224147926418976... A073226
3 e ↑↑ 3 eee 3814279.1047602205922... A073227
4 e ↑↑ 4 eeee 2.331504399... × 10 1656520 A085667