血气计算公式stirlings formula斯特林(stirling)公式

2020年10月10日发(作者：尤锦)
stirlings formula斯特林(stirling)公式

Stirling's FormulaAn important formula in applied
mathematics as well as in probability is the Stirling's formula
known as





where is used to indicate that the ratio of the two sides goes to
1 as n goes to . In other words, we have





or






Proof of the Stirling's Formula
First take the log of n! to get





Since the log function is increasing on the interval , we get





for . Add the above inequalities, with , we get





Though the first integral is improper, it is easy to show that in
fact it is convergent. Using the antiderivative of (being ), we
get





Next, set





We have





Easy algebraic manipulation gives





Using the Taylor expansion





for -1 < t < 1, we get





This implies





We recognize a geometric series. Therefore we have





From this we get

1.
the sequence is decreasing;
2.
the sequence is increasing. This will imply that converges to
a number C with





and that C > d1 - 112 = 1 - 112 = 1112. Taking the
exponential of dn, we get





The final step in the proof if to show that . This will be done via
Wallis formula (and Wallis integrals). Indeed, recall the limit





Rewriting this formula, we get





Playing with the numbers, we get





Using the above formula





we get





Easy algebra gives





since we are dealing with constants, we get in fact . This
completes the proof of the Stirling's formula.[Trigonometry]
[Calculus]
[Geometry] [Algebra] [Differential Equations]
[Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S.
Mathematics d A. Khamsi
Tue Dec 3 17:39:00 MST 1996