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Stirling's approximation

Asymptotic formulas and bounds for factorials and log-factorials for large n.
Stirling’s approximation

Stirling’s approximation gives accurate large-nn estimates for the factorial (factorial function) and related quantities such as binomial and multinomial coefficients.

Core asymptotic form

For integers n→∞n \to \infty,

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

A common refined statement is

n!=2πn(ne)n(1+O ⁣(1n)). n! = \sqrt{2\pi n}\left(\frac{n}{e}\right)^n\left(1+O\!\left(\frac1n\right)\right).

Logarithmic form

Taking logs (natural logarithm),

log⁡(n!)=nlog⁡n−n+12log⁡(2πn)+O ⁣(1n). \log(n!) = n\log n - n + \tfrac12\log(2\pi n) + O\!\left(\frac1n\right).

This form is often numerically stable and is the starting point for entropy-based approximations (see ).

Explicit error bounds (one standard version)

There are classical two-sided bounds of the form

2π nn+12e−n e112n+1  <  n!  <  2π nn+12e−n e112n. \sqrt{2\pi}\,n^{n+\frac12}e^{-n}\,e^{\frac{1}{12n+1}} \;<\; n! \;<\; \sqrt{2\pi}\,n^{n+\frac12}e^{-n}\,e^{\frac{1}{12n}}.

Equivalently, for n≥1n\ge 1,

112n+1  <  log⁡(n!)−(n+12)log⁡n+n−12log⁡(2π)  <  112n. \frac{1}{12n+1} \;<\; \log(n!) - \Bigl(n+\tfrac12\Bigr)\log n + n - \tfrac12\log(2\pi) \;<\; \frac{1}{12n}.

Asymptotic series refinement

Stirling’s approximation admits an asymptotic expansion in inverse powers of nn:

n!∼2πn(ne)n(1+112n+1288n2−13951840n3+⋯ ), n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1+\frac{1}{12n}+\frac{1}{288n^2}-\frac{139}{51840n^3}+\cdots\right),

which is useful for high-precision approximations (typically truncated after a few terms).

Typical use cases

  • Approximating combinatorial counts such as (nk)\binom{n}{k} and multinomial coefficients.
  • Converting counts into entropy-like expressions in information theory and large deviations (see ).
  • Approximating integrals via local quadratic expansions (conceptually similar to ).