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 Speed of convergence and error estimates 2 Derivation 3 History

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 .