List of representations of e

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.

As a continued fraction

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

$e = [2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots]. \,$

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

$e = [ 1 ; \textbf{0.5} , 12 , 5 , 28 , 9 , 44 , 13 , 60 , 17 , \ldots , \textbf{4(4n-1)} , \textbf{4n+1} , \ldots]. \,$

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

$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}} = 2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}$

$e = 2+\cfrac{1}{1+\cfrac{2}{5+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}} = 1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}}$

This last, equivalent to [1; 0.5, 12, 5, 28, 9, ...], is a special case of a general formula for the exponential function:

$e^{x/y} = 1+\cfrac{2x} {2y-x+\cfrac{x^2} {6y+\cfrac{x^2} {10y+\cfrac{x^2} {14y+\cfrac{x^2} {18y+\ddots}}}}}$

As an infinite series

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

$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$ for any real number x.

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

$e = \sum_{k=0}^\infty \frac{1}{k!}$, and

$e^{-1} = \sum_{k=0}^\infty \frac{(-1)^k}{k!}$

Other series include the following:

$e = \left [ \sum_{k=0}^\infty \frac{1-2k}{(2k)!} \right ]^{-1}$

$e = \frac{1}{2} \sum_{k=0}^\infty \frac{k+1}{k!}$

$e = 2 \sum_{k=0}^\infty \frac{k+1}{(2k+1)!}$

$e = \sum_{k=0}^\infty \frac{3-4k^2}{(2k+1)!}$

$e = \sum_{k=0}^\infty \frac{(3k)^2+1}{(3k)!}$

$e = \left [ \sum_{k=0}^\infty \frac{4k+3}{2^{2k+1}\,(2k+1)!} \right ]^2$

$e = \sum_{k=1}^\infty \frac{k^n}{B_n(k!)}$ where $B_n$ is the $n^{th}$ Bell number. Some few examples: (for n = 2..7)

$e = \sum_{k=1}^\infty \frac{k^2}{2(k!)}$

$e = \sum_{k=1}^\infty \frac{k^3}{5(k!)}$

$e = \sum_{k=1}^\infty \frac{k^4}{15(k!)}$

$e = \sum_{k=1}^\infty \frac{k^5}{52(k!)}$

$e = \sum_{k=1}^\infty \frac{k^6}{203(k!)}$

$e = \sum_{k=1}^\infty \frac{k^7}{877(k!)}$

Consideration of how to put upper bounds on e leads to this descending series:

$e = 3 + \sum_{k=2}^\infty \frac{-1}{k! (k-1) k} = 3 - \frac{1}{4} - \frac{1}{36} - \frac{1}{288} - \frac{1}{2400} - \frac{1}{21600} - \frac{1}{211680} - \frac{1}{2257920} - \cdots$

which gives at least one correct (or rounded up) digit per term. That is, if 1 ≤ n, then

$e < 3 + \sum_{k=2}^n \frac{-1}{k! (k-1) k} < e + 0.6 \cdot 10^{1-n} \,.$

As an infinite product

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

$e= 2 \left ( \frac{2}{1} \right )^{1/2} \left ( \frac{2}{3}\; \frac{4}{3} \right )^{1/4} \left ( \frac{4}{5}\; \frac{6}{5}\; \frac{6}{7}\; \frac{8}{7} \right )^{1/8} \cdots$

and Guillera's product

$e = \left ( \frac{2}{1} \right )^{1/1} \left (\frac{2^2}{1 \cdot 3} \right )^{1/2} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/3} \left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/4} \cdots ,$

where the nth factor is the nth root of the product

$\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}},$

as well as the infinite product

$e = \frac{2\cdot 2^{(\ln(2)-1)^2} \cdots}{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^3}\cdots }.$

More generally, if 1 < B < e² (which includes B = 2, 3, 4, 5, 6, or 7), then

$e = \frac{B\cdot B^{(\ln(B)-1)^2} \cdots}{B^{\ln(B)-1}\cdot B^{(\ln(B)-1)^3}\cdots }.$

As the limit of a sequence

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

$e= \lim_{n \to \infty} n\cdot\left ( \frac{\sqrt{2 \pi n}}{n!} \right )^{1/n}$ and

$e=\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}}$ (both by Stirling's formula).

The symmetric limit,

$e=\lim_{n \to \infty} \left [ \frac{(n+1)^{n+1}}{n^n}- \frac{n^n}{(n-1)^{n-1}} \right ]$

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

The next two definitions are direct corollaries of the prime number theorem

$e= \lim_{n \to \infty}(p_n \#)^{1/p_n}$

where $p_n$ is the nth prime and $p_n \#$ is the primorial of the nth prime.

$e= \lim_{n \to \infty}n^{\pi(n)/n}$

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

Also:

$e^x= \lim_{n \to \infty}\left (1+ \frac{x}{n} \right )^n.$

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

$e= \lim_{n \to \infty}\left (1+ \frac{1}{n} \right )^n.$

In trigonometry

Trigonometrically, e can be written as the sum of two hyperbolic functions:

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