Sequences, Series, and Special Functionsorli n! ( e )n 1
n-+1!, V'iim ;:; =
oras n ---+ oo. This result is known as Stirling's formula.
The asymptotic relation (8.20-27) can also be put in the formn! = (~)n ~eμ(n)where μ(n) ---+ 0 as n ---+ oo. From (8.20-28) we have
μ( n) = ln n! - ( n +^1 / 2 ) ln n + n - ln y'2;By taking logarithms in (8.20-25) we find that
n-1
lnn! - (n +^1 / 2 )lnn + n = 1-L cxk
k=l
so that
n-1
μ(n) = 1-2:::::cxk-lny'2;
. k=l
Letting n ---+ oo we have μ( n) ---+ 0, and we get
00
1-Lcxk = lny'2;
k=l
Therefore,
μ(n) = (1-I:~k)-(1-f ak)
k=l k=l
oo N
= ""CXk L..J = N lim -+oo ""CXk L..J
k=n k=nand making use of (8.20-21), we conclude that
625(~20-27)(8.20-28)(8.20-29)O<μ(n)<-^1 hm. (1 ----+···+----^1 1 1 ) =-^1
- 12 N-+oo n n+l N N+l 12n
(8.20-30)
H. Robbins [31] has shown that 1/(12n + 1) < μ(n) < 1/12n. See also
A. J. Maria [25] and W. Feller [12].
Graphs of y = r(x) and of v = if(z)I-For x real, r(x) is real with poles
at the points 0, -1, -2, .... Thus the lines x = 0, x = -1, x = -2, ... are