萨米大冒险全集:Stirling's Formula---斯特林(Stirling)公式
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Stirling's Formula
An 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.
and that C > d1 - 1/12 = 1 - 1/12 = 11/12. 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.
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996
Stirling's Formula---斯特林(Stirling)公式
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