List of representations of e

The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other sort of limit of a sequence.

As a continued fraction

Euler proved that the number e is represented as the infinite simple continued fraction[1] (sequence A003417 in the OEIS):

${\displaystyle e=[2;1,2,1,1,4,1,1,6,1,1,8,1,\ldots ,1,2n,1,\ldots ].}$

Its convergence can be tripled[clarification needed][citation needed] by allowing just one fractional number:

${\displaystyle e=[1;1/2,12,5,28,9,44,13,60,17,\ldots ,4(4n-1),4n+1,\ldots ].}$

Here are some infinite generalized continued fraction expansions of e. The second is generated from the first by a simple equivalence transformation.

${\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+{\cfrac {4}{5+\ddots }}}}}}}}}}=2+{\cfrac {2}{2+{\cfrac {3}{3+{\cfrac {4}{4+{\cfrac {5}{5+{\cfrac {6}{6+\ddots \,}}}}}}}}}}}$
${\displaystyle e=2+{\cfrac {1}{1+{\cfrac {2}{5+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{18+\ddots \,}}}}}}}}}}=1+{\cfrac {2}{1+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{18+\ddots \,}}}}}}}}}}}$

This last, equivalent to [1; 0.5, 12, 5, 28, 9, ...], is a special case of a general formula for the exponential function:

${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}}$

Conjectures

There are also continued fraction conjectures for e. For example, a computer program developed at the Israel Institute of Technology has came up with:[2]

${\displaystyle e=3+{\cfrac {-1}{4+{\cfrac {-2}{5+{\cfrac {-3}{6+{\cfrac {-4}{7+\ddots \,}}}}}}}}}$

in 2019. This conjecture has now been proved, according to [3]

As an infinite series

The number e can be expressed as the sum of the following infinite series:

${\displaystyle e^{x}=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}}$  for any real number x.

In the special case where x = 1 or −1, we have:

${\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}}$ ,[4] and
${\displaystyle e^{-1}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}.}$

Other series include the following:

${\displaystyle e=\left[\sum _{k=0}^{\infty }{\frac {1-2k}{(2k)!}}\right]^{-1}}$  [5]
${\displaystyle e={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {k+1}{k!}}}$
${\displaystyle e=2\sum _{k=0}^{\infty }{\frac {k+1}{(2k+1)!}}}$
${\displaystyle e=\sum _{k=0}^{\infty }{\frac {3-4k^{2}}{(2k+1)!}}}$
${\displaystyle e=\sum _{k=0}^{\infty }{\frac {(3k)^{2}+1}{(3k)!}}=\sum _{k=0}^{\infty }{\frac {(3k+1)^{2}+1}{(3k+1)!}}=\sum _{k=0}^{\infty }{\frac {(3k+2)^{2}+1}{(3k+2)!}}}$
${\displaystyle e=\left[\sum _{k=0}^{\infty }{\frac {4k+3}{2^{2k+1}\,(2k+1)!}}\right]^{2}}$
${\displaystyle e=\sum _{k=0}^{\infty }{\frac {k^{n}}{B_{n}(k!)}}}$  where ${\displaystyle B_{n}}$  is the nth Bell number.

Consideration of how to put upper bounds on e leads to this descending series:

${\displaystyle e=3-\sum _{k=2}^{\infty }{\frac {1}{k!(k-1)k}}=3-{\frac {1}{4}}-{\frac {1}{36}}-{\frac {1}{288}}-{\frac {1}{2400}}-{\frac {1}{21600}}-{\frac {1}{211680}}-{\frac {1}{2257920}}-\cdots }$

which gives at least one correct (or rounded up) digit per term. That is, if 1 ≤ n, then

${\displaystyle e<3-\sum _{k=2}^{n}{\frac {1}{k!(k-1)k}}

More generally, if x is not in {2, 3, 4, 5, ...}, then

${\displaystyle e^{x}={\frac {2+x}{2-x}}+\sum _{k=2}^{\infty }{\frac {-x^{k+1}}{k!(k-x)(k+1-x)}}\,.}$

As an infinite product

The number e is also given by several infinite product forms including Pippenger's product

${\displaystyle e=2\left({\frac {2}{1}}\right)^{1/2}\left({\frac {2}{3}}\;{\frac {4}{3}}\right)^{1/4}\left({\frac {4}{5}}\;{\frac {6}{5}}\;{\frac {6}{7}}\;{\frac {8}{7}}\right)^{1/8}\cdots }$

and Guillera's product [6][7]

${\displaystyle e=\left({\frac {2}{1}}\right)^{1/1}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/2}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/3}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/4}\cdots ,}$

where the nth factor is the nth root of the product

${\displaystyle \prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}},}$

as well as the infinite product

${\displaystyle e={\frac {2\cdot 2^{(\ln(2)-1)^{2}}\cdots }{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^{3}}\cdots }}.}$

More generally, if 1 < B < e2 (which includes B = 2, 3, 4, 5, 6, or 7), then

${\displaystyle e={\frac {B\cdot B^{(\ln(B)-1)^{2}}\cdots }{B^{\ln(B)-1}\cdot B^{(\ln(B)-1)^{3}}\cdots }}.}$

As the limit of a sequence

The number e is equal to the limit of several infinite sequences:

${\displaystyle e=\lim _{n\to \infty }n\cdot \left({\frac {\sqrt {2\pi n}}{n!}}\right)^{1/n}}$  and
${\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}}$  (both by Stirling's formula).

The symmetric limit,[8]

${\displaystyle e=\lim _{n\to \infty }\left[{\frac {(n+1)^{n+1}}{n^{n}}}-{\frac {n^{n}}{(n-1)^{n-1}}}\right]}$

may be obtained by manipulation of the basic limit definition of e.

The next two definitions are direct corollaries of the prime number theorem[9]

${\displaystyle e=\lim _{n\to \infty }(p_{n}\#)^{1/p_{n}}}$

where ${\displaystyle p_{n}}$  is the nth prime and ${\displaystyle p_{n}\#}$  is the primorial of the nth prime.

${\displaystyle e=\lim _{n\to \infty }n^{\pi (n)/n}}$

where ${\displaystyle \pi (n)}$  is the prime counting function.

Also:

${\displaystyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}$

In the special case that ${\displaystyle x=1}$ , the result is the famous statement:

${\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}$

The ratio of the factorial ${\displaystyle n!}$ , that counts all permutations of an orderet set S with cardinality ${\displaystyle n}$ , and the derangement function ${\displaystyle !n}$ , which counts the amount of permutations where no element appears in its original position, tends to ${\displaystyle e}$  as ${\displaystyle n}$  grows.

${\displaystyle e=\lim _{n\to \infty }{\frac {n!}{!n}}.}$

In trigonometry

Trigonometrically, e can be written in terms of the sum of two hyperbolic functions,

${\displaystyle e^{x}=\sinh(x)+\cosh(x),}$

at x = 1.

Notes

1. ^ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Retrieved 2017-04-23.
2. ^ Gal Raayoni; et al. (Jun 2019). "The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants". arXiv:1907.00205. Bibcode:2019arXiv190700205R. Missing or empty |url= (help)
3. ^ Gal Raayoni; et al. (Jun 2019). "The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants". arXiv:1907.00205. Bibcode:2019arXiv190700205R. Missing or empty |url= (help)
4. ^ Brown, Stan (2006-08-27). "It's the Law Too — the Laws of Logarithms". Oak Road Systems. Archived from the original on 2008-08-13. Retrieved 2008-08-14.
5. ^ Formulas 2–7: H. J. Brothers, Improving the convergence of Newton's series approximation for e, The College Mathematics Journal, Vol. 35, No. 1, (2004), pp. 34–39.
6. ^ J. Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729–734.
7. ^ J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent,Ramanujan Journal 16 (2008), 247–270.
8. ^ H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e, The Mathematical Intelligencer, Vol. 20, No. 4, (1998), pp. 25–29.
9. ^ S. M. Ruiz 1997