Stirling's Formula---斯特林(Stirling)公式

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.

This will imply that

converges to a number C with

 

 

and that C > d₁ - 1/12 = 1 - 1/12 = 11/12. Taking the exponential of d_n, 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.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996