Stirling's approximation

mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large factorials. It is named in honour of James Stirling. Formally, it states:

<math>\lim_{n \rightarrow \infty} {n!\over \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n} } = 1<math>

which is often written as

<math>n! \sim \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n}<math>

(See limit, square root, π, e.) For large n, the right hand side is a good approximation for n!, and much faster and easier to calculate. For example, the formula gives for 30! the approximation 2.6452 × 10³² while the correct value is about 2.6525 × 10³². The error is less than 0.3% in this case.

Speed of convergence and error estimates

More precisely,

<math>n! = \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n}e^{\lambda_n}<math>

with

<math>\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}.<math>

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):

<math>

As <math>n \to \infty<math>, the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion.

Derivation

The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!, one considers the natural logarithm ln(n!) = ln(1) + ln(2) + ... + ln(n); the Euler-Maclaurin formula gives estimates for sums like these. The goal, then, is to show the approximation formula in its logarithmic form:

<math>\ln n! \approx \left(n+\frac{1}{2}\right)\ln n - n +\ln\left(\sqrt{2\pi}\right)<math>

Alternatively, the leading term of Stirling's approximation can be obtained through the method of steepest descent.

History

The formula was first discovered by Abraham de Moivre in the form

<math>n!\sim [{\rm constant}]\cdot n^{n+1/2} e^{-n}<math>

Stirling's contribution consisted of showing that the "constant" is <math>\sqrt{2\pi}<math>.