# Stirling's approximation of factorial

Online calculator computes Stirling's approximation of factorial of given positive integer (up to 170!)

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#### Timur

Criado: 2011-06-15 12:36:32, Ultima atualização: 2021-03-03 07:43:03

Factorial n! of a positive integer n is defined as:
$n! = 1\cdot 2\cdot\ldots\cdot n =\prod_{i=1}^n i$
The special case 0! is defined to have value 0! = 1

There are several approximation formulae, for example, Stirling's approximation, which is defined as:
$n! = \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 + \frac{1}{12 n} + \frac{1}{288 n^2} - \frac{139}{51840 n^3}+O\left(n^{-4}\right)\right)$

For simplicity, only main member is computed
$n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$

with the claim that
$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n < n! < \sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{1/(12n)}$

This calculator computes factorial, then its approximation using Stirling's formula. Also, it computes lower and upper bounds from inequality above.

Unfortunately, because it operates with floating-point numbers to compute approximation, it has to rely on Javascript numbers and is limited to 170!

For the UNLIMITED factorial, check out this unlimited factorial calculator

#### Factorial

Digits after the decimal point: 2
n!

Lower bound

Stirling's approximation

Upper bound

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