List of representations of e

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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[edit]

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[edit]

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}{k!} = \sum_{k=1}^\infty \frac{1}{(k-1)!} = \sum_{k=0}^\infty \frac{1}{k!}
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[edit]

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[edit]

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[edit]

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

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

Notes[edit]

  1. ^ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?". MAA Online. Retrieved 2010-06-18. 
  2. ^ Brown, Stan (2006-08-27). "It’s the Law Too — the Laws of Logarithms". Oak Road Systems. Retrieved 2008-08-14. 
  3. ^ 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.
  4. ^ J. Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729–734.
  5. ^ 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.
  6. ^ 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.
  7. ^ Khattri, Sanjay. "From Lobatto Quadrature to the Euler constant e". 
  8. ^ S. M. Ruiz 1997