- 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::
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.:
:
:
Setting "m"="x" and "n"=2 yields
:
As an infinite series
The number "e" is also equal to the sum of the following
infinite series :: [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]
:
: [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.]
:
:
:
:
:
:
:
:
:
:
: where is the
Bell number .As an infinite product
The number "e" is also given by several
infinite product forms including Pippenger's product:
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.] :where the "n"th factor is the "n"th root of the product:
as well as the infinite product
:
As the limit of a sequence
The number "e" is equal to the limit of several
infinite sequences :: and
: (both by
Stirling's formula ).The symmetric limit,
: [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
: [ S. M. Ruiz 1997]
where is the "n"th prime and is the
primorial of the "n"th prime.Also::
And when the result is the famous statement:
:
Notes
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