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 OEIS|id=A003417:

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

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.

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

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

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

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:

:$e\; =\; sum\_\{k=0\}^infty\; frac\{1\}\{k!\}$ [cite web|url=http://oakroadsystems.com/math/loglaws.htm|title=It’s the Law Too — the Laws of Logarithms|last=Brown|first=Stan|date=2006-08-27|publisher=Oak Road Systems|accessdate=2008-08-14]

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

:$e\; =\; left\; [\; sum\_\{k=0\}^infty\; frac\{1-2k\}\{(2k)!\}\; ight\; ]\; ^\{-1\}$ [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; pages 34-39.]

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

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

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

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

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

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

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

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

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

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

:$e\; =\; sum\_\{k=1\}^infty\; frac\{k^n\}\{B\_n(k!)\}$ 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

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

and Guillera's product [ J. Sondow, A faster product for pi and a new integral for ln pi/2, "Amer. Math. Monthly" 112 (2005) 729-734.] :$e\; =\; left\; (\; frac\{2\}\{1\}\; ight\; )^\{1/1\}\; left\; (frac\{2^2\}\{1\; cdot\; 3\}\; ight\; )^\{1/2\}\; left\; (frac\{2^3\; cdot\; 4\}\{1\; cdot\; 3^3\}\; ight\; )^\{1/3\}\; left\; (frac\{2^4\; cdot\; 4^4\}\{1\; cdot\; 3^6\; cdot\; 5\}\; ight\; )^\{1/4\}\; cdots\; ,$where the "n"th factor is the "n"th root of the product:$prod\_\{k=0\}^n\; (k+1)^\{(-1)^\{k+1\}\{n\; choose\; k,$

as well as the infinite product

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

As the limit of a sequence

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

:$e=\; lim\_\{n\; o\; infty\}\; ncdotleft\; (\; frac\{sqrt\{2\; pi\; n\{n!\}\; ight\; )^\{1/n\}$ and

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

The symmetric limit,

:$e=lim\_\{n\; o\; infty\}\; left\; [\; frac\{(n+1)^\{n+1\{n^n\}-\; frac\{n^n\}\{(n-1)^\{n-1\; ight\; ]$ [H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e. "The Mathematical Intelligencer", Vol. 20, No. 4, 1998; pages 25-29.]

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

:$e=\; lim\_\{n\; o\; infty\}(p\_n\; \#)^\{1/p\_n\}$ [ S. M. Ruiz 1997]

where $p\_n$ is the "n"th prime and $p\_n\; \#$ is the primorial of the "n"th prime.

Also::$e^x=\; lim\_\{n\; o\; infty\}left\; (1+\; frac\{x\}\{n\}\; ight\; )^n$

And when $x\; =\; 1$ the result is the famous statement:

:$e=\; lim\_\{n\; o\; infty\}left\; (1+\; frac\{1\}\{n\}\; ight\; )^n$

Notes