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Factorials grow faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log(x)).
Just as triangular numbers sum the numbers from to , and factorials take their product, the exponential factorial exponentiates. The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is = These numbers grow much more quickly than regular factorials. [95] Falling factorial
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, [1] Edmund Landau, [2] and others, collectively called Bachmann–Landau notation or asymptotic notation.
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The number of digits in the exponential factorial of 6 is approximately 5 × 10 183 230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number:
A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:
Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth . In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration , and A ( n , n ) {\displaystyle A(n,n)} , the diagonal ...
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Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .