Stirling's approximation

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

One way of stating the approximation involves the logarithm of the factorial:

\ln(n!)=n\ln n-n+O(\ln n),

where the big O notation means that, for all sufficiently large values of $n$ , the difference between $\ln(n!)$ and $n\ln n-n$ will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form

\log _{2}(n!)=n\log _{2}n-n\log _{2}e+O(\log _{2}n).

The error term in either base can be expressed more precisely as ${\tfrac {1}{2}}\log(2\pi n)+O({\tfrac {1}{n}})$ , corresponding to an approximate formula for the factorial itself,

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

Here the sign $\sim$ means that the two quantities are asymptotic, that is, that their ratio tends to 1 as $n$ tends to infinity. The following version of the bound holds for all $n\geq 1$ , rather than only asymptotically:

{\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n+1}}<n!<{\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.

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