Stirling's approximation (or Stirling's formula) is an approximation for large factorials. It is named in honour of James Stirling. Formally, it states:
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2 Derivation 3 History |
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:
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:
The formula was first discovered by Abraham de Moivre in the form