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 (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, may also be represented as an infinite series, infinite product, or other types of limit of a sequence.

As a continued fraction

Euler proved that the number is represented as the infinite simple continued fraction :

:e = ; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, \ldots, 1, 2n, 1, \ldots

Its convergence can be tripled by allowing just one fractional number:

: e = ; 1/2, 12, 5, 28, 9, 44, 13, 60, 17, \ldots, 4(4n-1), 4n+1, \ldots

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

:$e= 2+\cfrac = 2+\cfrac$

:$e = 2+\cfrac = 1+\cfrac$

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

:$e^ = 1+\cfrac$

As an infinite series

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

:$e^x = \sum_^\infty \frac$ for any real number ''x''.

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

:$e = \sum_^\infty \frac$, and

:$e^ = \sum_^\infty \frac.$

Other series include the following:

:e = \left \sum_^\infty \frac \right

:$e = \frac \sum_^\infty \frac$

:$e = 2 \sum_^\infty \frac$

:$e = \sum_^\infty \frac$

:$e = \sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty \frac$

:e = \left \sum_^\infty \frac \right 2

:$e = \sum_^\infty \frac$ where $B_n$ is the th Bell number.

Consideration of how to put upper bounds on ''e'' leads to this descending series:
:$e = 3 - \sum_^\infty \frac = 3 - \frac - \frac - \frac - \frac - \frac - \frac - \frac - \cdots$
which gives at least one correct (or rounded up) digit per term. That is, if 1 ≤ ''n'', then
:$e < 3 - \sum_^n \frac < e + 0.6 \cdot 10^ \,.$
More generally, if ''x'' is not in , then
:$e^x = \frac + \sum_^\infty \frac \,.$

As an infinite product

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

:$e= 2 \left ( \frac \right )^ \left ( \frac\; \frac \right )^ \left ( \frac\; \frac\; \frac\; \frac \right )^ \cdots$

and Guillera's product
:$e = \left ( \frac \right )^ \left (\frac \right )^ \left (\frac \right )^ \left (\frac \right )^ \cdots ,$
where the ''n''th factor is the ''n''th root of the product
:$\prod_^n (k+1)^,$

as well as the infinite product

:$e = \frac.$

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

:$e = \frac.$

As the limit of a sequence

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

:$e= \lim_ n\cdot\left ( \frac \right )^$ and

:$e=\lim_ \frac$ (both by Stirling's formula).

The symmetric limit,

:e=\lim_ \left \frac- \frac \right /math>

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

The next two definitions are direct corollaries of the prime number theoremS. M. Ruiz 1997

:$e= \lim_(p_n \#)^$

where $p_n$ is the ''n''th prime and $p_n \#$ is the primorial of the ''n''th prime.

:$e= \lim_n^$

where $\pi(n)$ is the prime-counting function.

Also:
:$e^x= \lim_\left (1+ \frac \right )^n.$

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

:$e= \lim_\left (1+ \frac \right )^n.$

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

:$e= \lim_ \frac.$

In trigonometry

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

: $e^x = \sinh(x) + \cosh(x) ,$

at .

See also

* List of formulae involving π

Notes

{{DEFAULTSORT:e, Representations
Exponentials
Logarithms
E (mathematical constant)