List of representations of e
Part of a series of articles on the 
mathematical constant e 

Properties 
Applications 
Defining e 

People 
Related topics 
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.
Contents
As a continued fraction
Euler proved that the number e is represented as the infinite simple continued fraction^{[1]} (sequence A003417 in OEIS):
Its convergence can be tripled 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:
As an infinite series
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:
 ,^{[2]} and
Other series include the following:
 ^{[3]}
 where is the Bell number. Some few examples: (for n = 2..7)
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
As an infinite product
The number e is also given by several infinite product forms including Pippenger's product
and Guillera's product ^{[4]}^{[5]}
where the nth factor is the nth root of the product
as well as the infinite product
More generally, if 1 < B < e^{2} (which includes B = 2, 3, 4, 5, 6, or 7), then
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,^{[6]}^{[7]}
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]}
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:
In trigonometry
Trigonometrically, e can be written as the sum of two hyperbolic functions:
Notes
 ↑ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Retrieved 20100618.
 ↑ Brown, Stan (20060827). "It’s the Law Too — the Laws of Logarithms". Oak Road Systems. Retrieved 20080814.
 ↑ 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.
 ↑ J. Sondow, 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, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent,Ramanujan Journal 16 (2008), 247–270.
 ↑ H. J. Brothers and J. A. Knox, New closedform approximations to the Logarithmic Constant e, The Mathematical Intelligencer, Vol. 20, No. 4, (1998), pp. 25–29.
 ↑ Khattri, Sanjay. "From Lobatto Quadrature to the Euler constant e" (PDF).
 ↑ S. M. Ruiz 1997