Stirling's formula

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good quality approximation, leading to accurate results even for small values of $n$ . It is named after James Stirling, though it was first stated by Abraham de Moivre.

The formula as typically used in applications is

or, for instance in the worst-case lower bound for comparison sorting (by changing the base of the logarithm),

(in big O notation). The next term in the $O (ln n)$ is $1 / 2 ln(2π n)$ ; a more precise variant of the formula is therefore

where the sign ~ means that the two quantities are asymptotic, that is, their ratio tends to 1 as $n$ tends to infinity.

It is also possible to give a version of Stirling's formula with bounds valid for all positive integers $n$ , rather than asymptotics: one has

for all positive integers $n$ . Thus the ratio

is always between √2 $π$ = 2.5066... and $e$ = 2.71828....

The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating $n!$ , one considers its natural logarithm as this is a slowly varying function:

The right-hand side of this equation minus

is the approximation by the trapezoid rule of the integral

and the error in this approximation is given by the Euler–Maclaurin formula:

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