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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. 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!
 ,

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

See also:

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