- Representations of e
The
mathematical constant "e" can be represented in a variety of ways as areal number . Since "e" is anirrational number (seeproof that e is irrational ), it cannot be represented as a fraction, but it can be represented as acontinued fraction . Usingcalculus , "e" may also be represented as aninfinite series ,infinite product , or other sort oflimit 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
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