sformula斯特林公式

1、stirlings formula 斯特林 (stirling) 公式Stirlings FormulaAn important formula in applied mathematics as well as in probability is the Stirlings formula known as where is used to indicate that the ratio of the two sides goes to1 as n goes to . In other words, we have orProof of the Stirlings Formula First

2、 take the log of n! to getSince the log function is increasing on the interval , we get for . Add the above inequalities, with , we getThough the first integral is improper, it is easy to show that in fact it is convergent. Using the antiderivative of (being ), we getNext, setWe haveEasy algebraic m

3、anipulation givesUsing the Taylor expansion for -1 < t < 1, we getThis impliesWe recognize a geometric series. Therefore we haveFrom this we get1. the sequence is decreasing;2.converges tothe sequence is increasing. This will imply that a number C with and that C > d1 - 1/12 = 1 - 1/12 = 11

4、/12. Taking the exponential of dn, we getThe final step in the proof if to show that . This will be done via Wallis formula (and Wallis integrals). Indeed, recall the limitRewriting this formula, we getPlaying with the numbers, we getUsing the above formulawe getEasy algebra gives since we are dealing with constants, we get in fact . This completes the proof of the Stirlings formula.Trigonometry CalculusGeometry Algebra Differential EquationsComplex Variables Matrix AlgebraS.O.S MATHematics home pageDo you need more help? Please post your question on o