Top Qs
Timeline
Chat
Perspective

List of representations of e

From Wikipedia, the free encyclopedia

List of representations of e
Remove ads

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 types of limit of a sequence.

Remove ads

As a continued fraction

Summarize
Perspective

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

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

This last non-simple continued fraction (sequence A110185 in the OEIS), equivalent to , has a quicker convergence rate compared to Euler's continued fraction formula[clarification needed] and is a special case of a general formula for the exponential function:

Remove ads

As an infinite series

Summarize
Perspective

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 nth Bell number.
[4]

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

Remove ads

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[5] which equates to This fraction is of the form , where computes the sum of the terms from to .

Remove ads

As an infinite product

Summarize
Perspective

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

Also

Remove ads

As the limit of a sequence

Summarize
Perspective

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, is the primorial of the nth prime, and is the prime-counting function.

Also:

In the special case that x = 1, the result is the famous statement:

The ratio of the factorial n!, that counts all permutations of an ordered set S with cardinality n, and the subfactorial (a.k.a. the derangement function) !n, which counts the amount of permutations where no element appears in its original position, tends to e as n grows.

Remove ads

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 n trials tends to 1/e. That is,

Remove ads

As a binomial series

Summarize
Perspective

Consider the sequence:

By the binomial theorem:[10]

which converges to as increases. The term is the th falling factorial power of , which behaves like when is large. For fixed and as :

Remove ads

As a ratio of ratios

Summarize
Perspective

A unique representation of e 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 e. Specifically, the ratio of the products of binomial coefficients in adjacent rows of Pascal's triangle tends to e as the row number n 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.[11][12]

Remove ads

In trigonometry

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

at x = 1.

See also

Notes

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads