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Stirling formula

The French and Scottish mathematicians Abraham de Moivre and James Stirling studied the very rapid growth of factorials at the same period. In 1730, in his Miscellanea Analytica de seriebus et quadraturis De Moivre published the estimate

$large n!sim sqrt2pi nn^ne^-n" alt=" " />,

wrongly named Stirling's formula. This mathematical expression has now become part of popular culture, because it associates the famous numbers e and $pi$.

Stirling, who corresponded with de Moïvre, suggests an improvement to the formula published in his Methodus Differentialis sive Tractatus de Summatione et Interpolatione Serierum Infinitarum (1730):

$large n! = sqrt2pi nleft(fracneright)^n left( 1 + frac112n + frac1288n^2 - frac13951 840n^3 - frac5712 488 320n^4 +cdots right)" alt=" " />.

Note that this series is not convergent, despite the very good approximations it gives when truncated.

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