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Stirling's approximation of factorial

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

This page exists due to the efforts of the following people:

Timur

Timur

Created: 2011-06-15 12:36:32, Last updated: 2021-03-03 07:43:03
Creative Commons Attribution/Share-Alike License 3.0 (Unported)

This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/170/. Also, please do not modify any references to the original work (if any) contained in this content.

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

PLANETCALC, Factorial

Factorial

Digits after the decimal point: 2
n!
 
Lower bound
 
Stirling's approximation
 
Upper bound
 

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PLANETCALC, Stirling's approximation of factorial
Timur2021-03-03 07:43:03

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