List of representations of e

mathematical constant $e$
Part of a series of articles on the

Properties
Natural logarithm Exponential function
Applications
compound interest Euler's identity Euler's formula half-lives exponential growth and decay
Defining $e$
proof that $e$ is irrational representations of $e$ Lindemann–Weierstrass theorem
People
John Napier Leonhard Euler
Related topics
Schanuel's conjecture

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

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

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

Its convergence can be tripled by allowing just one fractional number:

e = [ 1 ; \textbf{0.5} , 12 , 5 , 28 , 9 , 44 , 13 , 60 , 17 , \ldots , \textbf{4(4n-1)} , \textbf{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.

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\,}}}}}

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:

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}}}}}

As an infinite series

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

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:

e = \sum_{k=0}^\infty \frac{1}{k!}

,^[2] and

e^{-1} = \sum_{k=0}^\infty \frac{(-1)^k}{k!}

Other series include the following:

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

^[3]

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 = \left [ -\frac{12}{\pi^2} \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^n}{B_n(k!)}

where

B_n

is the

n^{th}

Bell number. Some few examples: (for n=1,2,3)

e = \sum_{k=1}^\infty \frac{k^2}{2(k!)}

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^5}{52(k!)}

e = \sum_{k=1}^\infty \frac{k^6}{203(k!)}

e = \sum_{k=1}^\infty \frac{k^7}{877(k!)}

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 ^[4]^[5]

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,^[6]^[7]

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^[8]

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.

e= \lim_{n \to \infty}n^{\pi(n)/n}

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

Also:

e^x= \lim_{n \to \infty}\left (1+ \frac{x}{n} \right )^n.

In the special case that $x = 1$ , the result is the famous statement:

e= \lim_{n \to \infty}\left (1+ \frac{1}{n} \right )^n.

In trigonometry

Trigonometrically, $e$ can be written as the sum of two hyperbolic functions:

e^x = \sinh(x) + \cosh(x)\,

Notes

↑ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Retrieved 2010-06-18.
↑ Brown, Stan (2006-08-27). "It’s the Law Too — the Laws of Logarithms". Oak Road Systems. Retrieved 2008-08-14.
↑ 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.
↑ J. Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729–734.
↑ 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.
↑ 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.
↑ Khattri, Sanjay. "From Lobatto Quadrature to the Euler constant e" (PDF).
↑ S. M. Ruiz 1997

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