2—Infinite Series 41
2.6 Stirling’s Approximation
The Gamma function for positive integers is a factorial. A clever use of infinite series and Gaussian integrals
provides a useful approximate value for the factorial of largen.
n!∼
√
2 πnnne−n for largen (15)
Start from the Gamma function ofn+ 1.
n! = Γ(n+ 1) =
∫∞
0
dttne−t=
∫∞
0
dte−t+nlnt
The integrand starts at zero, increases, and drops back down to zero ast→∞. The graph roughly resembles a
Gaussian, and I can make this more precise by expanding the exponent around the point where it is a maximum.
The largest contribution to the whole integral comes from the region near this point. Differentiate the exponent
to find the maximum:
t
t=n= 5
t^5 e−t
21. 06
d
dt
(
−t+nlnt
)
=−1 +
n
t
= 0 gives t=n
Expand about this point
f(t) =−t+nlnt=f(n) + (t−n)f′(n) + (t−n)^2 f′′(n)/2 +···
=−n+nlnn+ 0 + (t−n)^2 (−n/n^2 )/2 +···
Keep terms only to the second order and the integral is approximately
n!∼
∫∞
0
dte−n+nlnn−(t−n)
(^2) / 2 n
=nne−n
∫∞
0
dte−(t−n)
(^2) / 2 n
(16)