Stirling's approximation

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

which is often written as

(See limit, square root, &pi, 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 2.6451 × 10³² while the correct value is about 2.6525 × 10³².

Table of contents

1 Consequences 2 Speed of convergence and error estimates 3 Derivation 4 History

Consequences

It can be shown that

using Stirling's appoximation.

Speed of convergence and error estimates

The speed of convergence of the above limit is expressed by the formula

where Θ(1/n) denotes a function whose asymptotical behavior for n→∞ is like a constant times 1/n; see Big O notation.

More precisely still:

with

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:

History

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

Stirling's contribution consisted of showing that the "constant" is .