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

*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 × 10

^{32}while the correct value is about 2.6525 × 10

^{32}.

Table of contents |

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:

## 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