List of representations of e

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 the quotient of two integers, 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 fractionEdit

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

 

Its convergence can be tripled[clarification needed][citation needed] by allowing just one fractional number:

 

Here are some infinite generalized continued fraction expansions of e. The second is generated from the first by a simple equivalence transformation.

 
 

This last, equivalent to [1; 0.5, 12, 5, 28, 9, ...], is a special case of a general formula for the exponential function:

 

ConjecturesEdit

There are also continued fraction conjectures for e. For example, a computer program developed at the Israel Institute of Technology has came up with:[2]

 

in 2019. This conjecture has now been proved, according to [3]

As an infinite seriesEdit

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

  for any real number x.

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

 ,[4] and
 

Other series include the following:

  [5]
 
 
 
 
 
  where   is the nth Bell number.

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

 

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

 

As an infinite productEdit

The number e is also given by several infinite product forms including Pippenger's product

 

and Guillera's product [6][7]

 

where the nth factor is the nth root of the product

 

as well as the infinite product

 

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

 

As the limit of a sequenceEdit

The number e is equal to the limit of several infinite sequences:

  and
  (both by Stirling's formula).

The symmetric limit,[8]

 

may be obtained by manipulation of the basic limit definition of e.

The next two definitions are direct corollaries of the prime number theorem[9]

 

where   is the nth prime and   is the primorial of the nth prime.

 

where   is the prime counting function.

Also:

 

In the special case that  , the result is the famous statement:

 

The ratio of the factorial  , that counts all permutations of an orderet set S with cardinality  , and the derangement function  , which counts the amount of permutations where no element appears in its original position, tends to   as   grows.

 

In trigonometryEdit

Trigonometrically, e can be written in terms of the sum of two hyperbolic functions,

 

at x = 1.

NotesEdit

  1. ^ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Retrieved 2017-04-23.
  2. ^ Gal Raayoni; et al. (Jun 2019). "The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants". arXiv:1907.00205. Bibcode:2019arXiv190700205R. Missing or empty |url= (help)
  3. ^ Gal Raayoni; et al. (Jun 2019). "The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants". arXiv:1907.00205. Bibcode:2019arXiv190700205R. Missing or empty |url= (help)
  4. ^ Brown, Stan (2006-08-27). "It's the Law Too — the Laws of Logarithms". Oak Road Systems. Archived from the original on 2008-08-13. Retrieved 2008-08-14.
  5. ^ 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.
  6. ^ J. Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729–734.
  7. ^ 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.
  8. ^ 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.
  9. ^ S. M. Ruiz 1997