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Euler's number

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Euler’s number
e
, sometimes called Napier’s constant, is the base of the exponential function and the natural logarithm. The number
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!
 ,
e  =  (1 + x) 1 / x  =  1 +
1
x
  x,
e  = 
nn
n!
 1 / n  = 
n
n  n! 
 ,

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
   =  0 + 
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 = e  ^  e
15.15426224147926418976... A073226
3
e ↑↑ 3
eee = e  ^  e  ^  e
3814279.1047602205922... A073227
4
e ↑↑ 4
eeee = e  ^  e  ^  e  ^  e
2.331504399... × 10 1656520 A085667
A004002 Benford numbers:
a (n) = e  ^  e  ^    ^  e
(
n
times,
n   ≥   0
) rounded to nearest integer.
{1, 3, 15, 3814279, ...}

See also

Notes

External links