# Stirling's Formula

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## Stirling's formula

[′stir·liŋz ‚fȯr·myə·lə]*n*/

*e*)

^{n}√(2π

*n*) is asymptotic to factorial

*n;*that is, the limit as

*n*goes to ∞ of their ratio is 1.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Stirling’s Formula

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 *e ^{wln} –* 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.

### REFERENCE

Fikhtengol’ts, G. M.*Kurs differentsial’nogo i integral’nogo ischisleniia*, 7th ed., vol. 2. Moscow, 1969.