List of representations of e
The mathematical constant can be represented in a variety of ways as a real number. Since is an irrational number, it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
As a continued fraction
proved that the number is represented as the infinite simple continued fraction :Here are some infinite generalized continued fraction expansions of. The second is generated from the first by a simple equivalence transformation.
This last non-simple continued fraction, equivalent to, has a quicker convergence rate compared to Euler's continued fraction formula and is a special case of a general formula for the exponential function:
As an infinite series
The number can be expressed as the sum of the following infinite series:In the special case where or, we have:
Other series include the following:
Consideration of how to put upper bounds on leads to this descending series:
which gives at least one correct digit per term. That is, if, then
More generally, if is not in, then
As a recursive function
The [|series representation] of, given as can also be expressed using a form of recursion. When is iteratively factored from the original series the result is the nested series which equates to This fraction is of the form, where computes the sum of the terms from to.As an infinite product
The number is also given by several infinite product forms including Pippenger's productand Guillera's product
where the th factor is the th root of the product
as well as the infinite product
More generally, if , then
Also
As the limit of a sequence
The number is equal to the limit of several infinite sequences:The symmetric limit,
may be obtained by manipulation of the basic limit definition of.
The next two definitions are direct corollaries of the prime number theorem
where is the th prime, is the primorial of the th prime, and 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 ordered set with cardinality, and the subfactorial , which counts the amount of permutations where no element appears in its original position, tends to as grows.
As a limiting probability
If we consider an event which has a probability of of occurring in any one trial, then the probability of the event not occurring in trials tends to.That is,
As a [binomial series]
Consider the sequence:By the binomial theorem:
which converges to as increases. The term is the th falling factorial power of, which behaves like when is large. For fixed and as :
As a ratio of ratios
A unique representation of can be found within the structure of Pascal's triangle, as discovered by Harlan Brothers. Pascal's triangle is composed of binomial coefficients, which are traditionally summed to derive polynomial expansions. However, Brothers identified a product-based relationship between these coefficients that links to. Specifically, the ratio of the products of binomial coefficients in adjacent rows of Pascal's triangle tends to as the row number increases:For.
The details of this relationship and its proof are outlined in the discussion on the properties of the rows of Pascal's triangle.
In trigonometry
Trigonometrically, can be written in terms of the sum of two hyperbolic functions,at.