Stirling's approximation of factorial

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

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Created: 2011-06-15 12:36:32, Last updated: 2021-03-03 07:43:03
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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



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

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PLANETCALC, Stirling's approximation of factorial