List of representations of e

Euler proved that the number {{math|e}} is represented as the infinite simple continued fraction{{cite web|url=http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf|title=How Euler Did It: Who proved e is Irrational?|last=Sandifer|first=Ed|date=Feb 2006|publisher=MAA Online|accessdate=2017-04-23}} {{OEIS|id=A003417}}:

:$$

\begin{align}

e & = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, \ldots, 1, 2n, 1, \ldots] \\[8pt]

& = 2 + \cfrac1{1 + \cfrac1{2 + \cfrac1{1 + \cfrac1{1 + \cfrac1{4 + \cfrac1{1 + \cfrac1{1 + \cfrac1{6 + \cfrac1{1 + \cfrac1{1 + \cfrac1{8 + {{} \atop \ddots}}}}}}} }}}}}

\end{align}

Here are some infinite generalized continued fraction expansions of {{math|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+{{}\atop\ddots}}}}}} = 2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+{{}\atop\ddots}\,}}}}}

:$e\; =\; 2+\backslash cfrac\{1\}\{1+\backslash cfrac\{2\}\{5+\backslash cfrac\{1\}\{10+\backslash cfrac\{1\}\{14+\backslash cfrac\{1\}\{18+\{\{\}\backslash atop\backslash ddots\}\backslash ,\}\}\}\}\}\; =\; 1+\backslash cfrac\{2\}\{1+\backslash cfrac\{1\}\{6+\backslash cfrac\{1\}\{10+\backslash cfrac\{1\}\{14+\backslash cfrac\{1\}\{18+\{\{\}\backslash atop\backslash ddots\}\backslash ,\}\}\}\}\}$

This last non-simple continued fraction {{OEIS|A110185}}, equivalent to $e\; =\; [1;\; 0.5,\; 12,\; 5,\; 28,\; 9,\; ...]$, has a quicker convergence rate compared to Euler's continued fraction formula{{clarify|reason=By what measure?|date = April 2017}} and is a special case of a general formula for the exponential function:

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

The number {{math|e}} can be expressed as the sum of the following infinite series:

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

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

:$e\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}$,{{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|url-status=usurped|archiveurl=https://web.archive.org/web/20080813175402/http://oakroadsystems.com/math/loglaws.htm|archivedate=2008-08-13}} and

:$e^\{-1\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(-1)^k\}\{k!\}.$

Other series include the following:

:$e\; =\; \backslash left\; [\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1-2k\}\{(2k)!\}\; \backslash right\; ]^\{-1\}$ Formulas 2–7: H. J. Brothers, [http://www.brotherstechnology.com/docs/Improving_Convergence_(CMJ-2004-01).pdf Improving the convergence of Newton's series approximation for e], The College Mathematics Journal, Vol. 35, No. 1, (2004), pp. 34–39.

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

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

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

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

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

:$e\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{k^n\}\{B\_n(k!)\}$ where $B\_n$ is the {{mvar|n}}th Bell number.

:$e\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{2k+3\}\{(k+2)!\}$Formula 8: A. G. Llorente, [https://osf.io/3yzbj A Novel Simple Representation Series for Euler's Number e], preprint, 2023.

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

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

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

:

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

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

The series representation of $e$, given as $$e\; =\; \backslash frac\{1\}\{0!\}\; +\; \backslash frac\{1\}\{1!\}\; +\; \backslash frac\{1\}\{2!\}\; +\; \backslash frac\{1\}\{3!\}\; +\; \backslash cdots$$can also be expressed using a form of recursion. When $\backslash textstyle\; \backslash frac\{1\}\{n\}$ is iteratively factored from the original series the result is the nested series{{citation

| journal = Wolfram MathWorld

| title = e

| pages = ex. 17, 18, and 19

| url = https://mathworld.wolfram.com/e.html

| archive-url = https://web.archive.org/web/20230315161656/https://mathworld.wolfram.com/e.html

| archive-date = 2023-03-15

}}. $$e\; =\; 1\; +\; \backslash frac\{1\}\{1\}\backslash left(1\; +\; \backslash frac\{1\}\{2\}\backslash left(1\; +\; \backslash frac\{1\}\{3\}\backslash left(1\; +\; \backslash cdots\; \backslash right)\backslash right)\backslash right)$$which equates to $$e\; =\; 1\; +\; \backslash cfrac\{1\; +\; \backslash cfrac\{1\; +\; \backslash cfrac\{1\; +\; \backslash cdots\; \}\{3\}\}\{2\}\}\{1\}$$ This fraction is of the form $\backslash textstyle\; f(n)\; =\; 1\; +\; \backslash frac\{f(n\; +\; 1)\}\{n\}$, where $f(1)$ computes the sum of the terms from $1$ to $\backslash infty$.

The number {{math|e}} is also given by several infinite product forms including Pippenger's product

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

and Guillera's product J. Sondow, [https://arxiv.org/abs/math/0401406 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, [https://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent], Ramanujan Journal 16 (2008), 247–270.

:$e\; =\; \backslash left\; (\; \backslash frac\{2\}\{1\}\; \backslash right\; )^\{1/1\}\; \backslash left\; (\backslash frac\{2^2\}\{1\; \backslash cdot\; 3\}\; \backslash right\; )^\{1/2\}\; \backslash left\; (\backslash frac\{2^3\; \backslash cdot\; 4\}\{1\; \backslash cdot\; 3^3\}\; \backslash right\; )^\{1/3\}$

\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

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

as well as the infinite product

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

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

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

Also

:$e\; =\; \backslash lim\backslash limits\_\{n\backslash rightarrow\backslash infty\}\backslash prod\_\{k=0\}^n\{n\; \backslash choose\; k\}^\{2/\{((n$

+\alpha)(n+\beta))}}\ \forall\alpha,\beta\in\Bbb R

The number {{math|e}} is equal to the limit of several infinite sequences:

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

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

The symmetric limit,H. J. Brothers and J. A. Knox, [http://www.brotherstechnology.com/docs/Closed-Form_Approximations_(MI-1998-12).pdf New closed-form approximations to the Logarithmic Constant e], The Mathematical Intelligencer, Vol. 20, No. 4, (1998), pp. 25–29.

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

may be obtained by manipulation of the basic limit definition of {{math|e}}.

The next two definitions are direct corollaries of the prime number theorem{{Cite journal |last=Ruiz |first=Sebastian Martin |title=81.27 A result on prime numbers |journal=The Mathematical Gazette |year=1997 |volume=81 |issue=491 |page=269|publisher=Cambridge University Press |url=https://doi.org/10.2307/3619207 |doi=10.2307/3619207|url-access=subscription }}

:$$

\begin{align}

e &= \lim_{n \to \infty}(p_n \#)^{1/p_n} \\

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

&= \lim_{n \to \infty} n^{n/p_n}

\end{align}

where $p\_n$ is the nth prime, $p\_n\; \backslash \#$ is the primorial of the nth prime, and $\backslash pi(n)$ is the prime-counting function.

Also:

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

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

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

The ratio of the factorial $n!$, that counts all permutations of an ordered set S with cardinality $n$, and the subfactorial (a.k.a. the derangement function) $!n$, which counts the amount of permutations where no element appears in its original position, tends to $e$ as $n$ grows.

:$e=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{n!\}\{!n\}.$

If we consider an event which has a probability of $\backslash frac\{1\}\{n\}$ of occurring in any one trial, then the probability of the event not occurring in n trials tends to $1\; /\; e$.

That is, $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\backslash left\; (1\; -\; \backslash frac\{1\}\{n\}\; \backslash right\; )^n\; =\; \backslash frac\{1\}\{e\}$

Consider the sequence:

:$$

e_n = \left(1 + \frac{1}{n}\right)^n

By the binomial theorem:{{cite book

| last=Stewart

| first=James

| title=Calculus: Early Transcendentals

| edition=6th

| publisher=Brooks/Cole Cengage Learning

| year=2008

| page=742

|url=https://archive.org/details/calculusearlytra0000jame/page/742/mode/2up

}}

:$e\_n\; =\; \backslash sum\_\{k=0\}^n\; \{n\; \backslash choose\; k\}\; \backslash frac\{1\}\{n^k\}\; =\; \backslash sum\_\{k=0\}^\{n\}\; \backslash frac\{n^\{\backslash underline\{k\}\}\}\{k!\}\; \backslash frac\{1\}\{n^k\}$

which converges to $e$ as $n$ increases. The term $n^\{\backslash underline\{k\}\}$ is the $k$th falling factorial power of $n$, which behaves like $n^k$ when $n$ is large. For fixed $k$ and as $\backslash textstyle\; n\; \backslash to\; \backslash infty$:

:$$

\frac{n^{\underline{k}}}{n^k} \approx 1 - \frac{k(k - 1)}{2n}

A unique representation of {{math|e}} can be found within the structure of Pascal's triangle, as discovered by Harlan Brothers. Pascal's triangle is composed of binomial coefficients, which are traditionally summed to derive polynomial expansions. However, Brothers identified a product-based relationship between these coefficients that links to {{math|e}}. Specifically, the ratio of the products of binomial coefficients in adjacent rows of Pascal's triangle tends to {{math|e}} as the row number {{math|n}} increases:

:$$

\begin{align}

P(n) &= \sum_{k = 0}^{n} \ln \binom{n}{k} \\

A &= P(n - 1), B = P(n), C = P(n + 1) \\

x &= (A - B) + (C - B) \sim 1 \\

\end{align}

:$\backslash exp\; x\; \backslash sim\; e$

The details of this relationship and its proof are outlined in the discussion on the properties of the rows of Pascal's triangle.{{Cite journal |last=Brothers |first=Harlan |title=Pascal's Triangle: The Hidden Stor-e |journal=The Mathematical Gazette |volume=96 |pages=145–148 |year=2012 |doi=10.1017/S0025557200004204}}{{Cite journal |last=Brothers |first=Harlan |title=Math Bite: Finding e in Pascal's Triangle |journal=Mathematics Magazine |volume=85 |issue=1 |pages=51 |year=2012 |doi=10.4169/math.mag.85.1.51}}

Trigonometrically, {{math|e}} can be written in terms of the sum of two hyperbolic functions,

: $e^x\; =\; \backslash sinh(x)\; +\; \backslash cosh(x)\; ,$

at {{math|1=x = 1}}.

• List of formulae involving π

{{reflist}}

{{DEFAULTSORT:e, Representations}}

Category:Exponentials

Category:Logarithms

Category:E (mathematical constant)