Example

Tag 08G7, permanent. Cite it as [Bourbaki, Tag 08G7].

Functions of a Real Variable, chapter V, § 4. APPLICATION TO SERIES WITH POSITIVE TERMS

Stirling’s formula. Let us seek an asymptotic expansion for $n!$ ; this leads us to expand $s_n = \sum_{p=1}^{n} \log p$ , and then $\exp(s_n)$ . The method of n 2 gives successively

s_n = \sum_{p=1}^{n} \log p \sim \int_{1}^{n} \log t \, dt = n \log n - n + 1

then

\log n - \int_{n-1}^{n} \log t \, dt = \log n - (n \log n - (n-1) \log(n-1) - 1) \sim \frac{1}{2n}

whence

s_n = n \log n - n + \frac{1}{2} \log n + o(\log n).

Then

\log n - \int_{n-1}^{n} \log t \, dt - \frac{1}{2} (\log n - \log(n-1)) \sim -\frac{1}{12n^2}

whence

s_n = n \log n - n + \frac{1}{2} \log n + k + \frac{1}{12n} + o_1 \left( \frac{1}{n} \right) \quad (k \text{ constant})

and one finally deduces (V, p. 226)

n! = e^k \, n^{n+1/2} \, e^{-n} \left( 1 + \frac{1}{12n} + o_2 \left( \frac{1}{n} \right) \right). \tag{5}

We shall show in VII, p. 322, that

e^k = \sqrt{2\pi}

. The formula (5) (with this value of

k

) is called Stirling's formula. In the same way, for every real number

a

not an integer

> 0

, one shows that

(a+1)(a+2)\ldots(a+n) \sim K(a) \, n^{n+a+\frac{1}{2}} \, e^{-n}. \tag{6}

We shall determine the function

K(a)

too (VII, p. 18). From formulae (5) and (6) one derives in particular that

\binom{a}{n} \sim (-1)^n \varphi(a) \, n^{-a-1} \tag{7}

for every real number

a

not an integer

> 0

, where

\varphi(a)

is a function of

a

which will be specified in VII, p. 322.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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