Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 567 — #575
14.5. Products 567
We can then apply our summing tools to find a closed form (or approximate closed
form) for ln.P / and then exponentiate at the end to undo the logarithm.
For example, let’s see how this works for the factorial function nŠ. We start by
taking the logarithm:
ln.nŠ/ D ln.1 2 3 .n 1/ n/
D ln.1/ C ln.2/ C ln.3/ C C ln.n
nX
D ln.i/:
iD1
1/ C ln.n/
Unfortunately, no closed form for this sum is known. However, we can apply
Theorem 14.3.2 to find good closed-form bounds on the sum. To do this, we first
compute
Z n
1
ln.x/ dx D x ln.x/
Plugging into Theorem 14.3.2, this means that
ˇ
x
ˇn
1
D n ln.n/ n C 1:
n ln.n/
Exponentiating then gives
n C 1
nX
ln.i/ n ln.n/
iD1
n C 1 C ln.n/:
This means that nŠ is within a factor of n of n n =e n 1 .
14.5.1 Stirling’s Formula
n n
nnC1
nŠ
en 1 e n 1 : (14.23)
The most commonly used product in discrete mathematics is probably nŠ, and
mathematicians have workedto find tight closed-form bounds on its value. The
most useful bounds are given in Theorem 14.5.1.
Theorem 14.5.1 (Stirling’s Formula). For all n 1,
nŠ D p n
n
2n e
.n/
e
where
1
12n C 1 .n/ 1
12n :