List of representations of e
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|mathematical constant 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
Its convergence can be tripled by allowing just one fractional number:
This last, equivalent to [1; 0.5, 12, 5, 28, 9, ...], is a special case of a general formula for the exponential function:
As an infinite series
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:
- , and
Other series include the following:
- where is the Bell number. Some few examples: (for n = 2..7)
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
As an infinite product
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
As the limit of a sequence
- (both by Stirling's formula).
may be obtained by manipulation of the basic limit definition of e.
where is the prime counting function.
In the special case that , the result is the famous statement:
Trigonometrically, e can be written as the sum of two hyperbolic functions:
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