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Stirling's formula[′stir·liŋz ‚fȯr·myə·lə]
a formula giving the approximate value of the product of the first n natural numbers 1 × 2 × . . . × n = n! when n is large. In other words, the formula provides an approximation of the factorial of n. Stirling’s formula was discovered by J. Stirling, who published it in 1730. He did not, however, provide an estimate of the error. The formula establishes the approximate equality
Here, π = 3.14159 . . ., and e = 2.71828 ... is the base of the natural logarithms.
When n! is calculated by means of this formula, the relative error is less than ewln – 1 and thus approaches 0 as n increases without bound. When n = 10, for example, the formula yields n! = 3,598,700, whereas the exact value of 10! is 3,628,800. In this case, the relative error is less than 1 percent.
Stirling’s formula has numerous applications in probability theory and mathematical statistics.