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 a fraction, 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

The number e can be represented as an infinite simple continued fraction (sequence A003417 in the OEIS):

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

Here are some infinite generalized continued fraction expansions of e. The second of these can be generated from the first by a simple equivalence transformation. The third one – with ... 6, 10, 14, ... in it – converges very quickly.

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

$e=1+{\cfrac {2}{1+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{\ddots \,}}}}}}}}}}$

$e^{2m/n}=1+{\cfrac {2m}{(n-m)+{\cfrac {m^{2}}{3n+{\cfrac {m^{2}}{5n+{\cfrac {m^{2}}{7n+{\cfrac {m^{2}}{\ddots \,}}}}}}}}}}$

Setting m=x and n=2 yields

$e^{x}=1+{\cfrac {2x}{(2-x)+{\cfrac {x^{2}}{6+{\cfrac {x^{2}}{10+{\cfrac {x^{2}}{14+{\cfrac {x^{2}}{\ddots \,}}}}}}}}}}$

As an infinite series

The number e is also equal to the sum of the following infinite series:

$e=\left[\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}\right]^{-1}$

$e=\left[\sum _{k=0}^{\infty }{\frac {1-2k}{(2k)!}}\right]^{-1}$

$e={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {k+1}{k!}}$

$e=2\sum _{k=0}^{\infty }{\frac {k+1}{(2k+1)!}}$

$e=\sum _{k=0}^{\infty }{\frac {3-4k^{2}}{(2k+1)!}}$

$e=\sum _{k=0}^{\infty }{\frac {(3k)^{2}+1}{(3k)!}}$

$e=\left[\sum _{k=0}^{\infty }{\frac {4k+3}{2^{2k+1}\,(2k+1)!}}\right]^{2}$

$e=-{\frac {12}{\pi ^{2}}}\left[\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\ \cos \left({\frac {9}{k\pi +{\sqrt {k^{2}\pi ^{2}-9}}}}\right)\right]^{-1/3}$

$e=\sum _{k=1}^{\infty }{\frac {k^{2}}{2(k!)}}$

$e=\sum _{k=1}^{\infty }{\frac {k}{2(k-1)!}}$

$e=\sum _{k=1}^{\infty }{\frac {k^{3}}{5(k!)}}$

$e=\sum _{k=1}^{\infty }{\frac {k^{4}}{15(k!)}}$

$e=\sum _{k=1}^{\infty }{\frac {k^{n}}{B_{n}(k!)}}$ where $B_{n}$ is the $n^{th}$ Bell number.

As an infinite product

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

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

and Guillera's product

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

where the nth factor is the nth root of the product

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

as well as the infinite product

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

As the limit of a sequence

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

$e=\lim _{n\to \infty }n\cdot \left({\frac {\sqrt {2\pi n}}{n!}}\right)^{1/n}$ and

$e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}$ (both by Stirling's formula).

The symmetric limit,

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

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

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

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

Finally, the famous sentence:

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