List of representations of e


The mathematical constant E | 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 sort of limit of a sequence.

As a continued fraction

proved that the number is represented as the infinite simple continued fraction :
Its convergence can be tripled by allowing just one fractional number:
Here are some infinite generalized continued fraction expansions of. The second is generated from the first by a simple equivalence transformation.
This last, equivalent to , is a special case of a general formula for the exponential function:

Conjectures

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

As an infinite series

The number can be expressed as the sum of the following infinite series:
In the special case where x = 1 or −1, we have:
Other series include the following:
Consideration of how to put upper bounds on e leads to this descending series:
which gives at least one correct digit per term. That is, if 1 ≤ n, then
More generally, if x is not in, then

As an infinite product

The number is also given by several infinite product forms including Pippenger's product
and Guillera's product
where the nth factor is the nth root of the product
as well as the infinite product
More generally, if 1 < B < e2, then

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 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, can be written in terms of the sum of two hyperbolic functions,
at.