Stirling's approximation

A good estimate of a factorial; the relative error shrinks as n grows. History: 1730.

n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n

How to work it out

1
The statement:
$n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$
2
A good estimate of a factorial; the relative error shrinks as n grows.
3
Below, the numbers in the table are computed by this site from the definition, not copied.

LaTeX source of the formulas on this page

\[ n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \]

n	n!	Stirling	Ratio
1	1	0.92213701	0.92214
5	120	118.01917	0.98349
10	3628800	3598695.6	0.99170
20	2432902008176640000	2422786800000000000	0.99584
50	≈ 64.48 (log10)	≈ 64.48 (log10)	0.99833

What is the stirling's approximation?

A good estimate of a factorial; the relative error shrinks as n grows.

Who is the Stirling's approximation associated with?

Leonhard Euler.