Remember **N! = N (N-1) (N-2) ... 3 x 2 x 1**? Stirling
came up with a great formula:

This formula is about 8% wrong for **1!**, and 0.8% wrong
for **10!** - it gives better and better approximations for
**N!** as **N** gets bigger and bigger. In
fact, the fractional error becomes really close to **1/(12 N)**:
if you multiply Stirling's formula by **1+1/(12 N)** you get
a formula which gets better and better even faster (with errors going
as **1/N ^{2}**).
Bender and Orszag give an even
better formula, involving

This is typical of *asymptotic series*. An
asymptotic series of
length **n** approaches **f(z)** as
**z** gets big, but for fixed **z** it can
diverge as **n** gets larger and larger. This is still
useful, but it isn't as stringent as the criteria for *convergent
series*, which have to approach for each **z** as
**n** gets large (as described, for example, in
Matthews and Walker).

You might have noticed that we slipped something in back there. What
is 0! (zero factorial), anyway? Well, you know that
**N! = N * (N-1)!**, so **(N-1)! = N! / N**,
and so 0! = 1. (Try it: 3! = 3 x 2 x 1 = 6; 2! = 2 = 3!/3, 1! = 1 = 2!/2,
so 0! = 1!/1 = 1). Taking this further, we find out that (-1)! = 1/0 =
infinity, (-2)! = (-1)! / -1 = -infinity,
(-3)! = (-2)!/(-2) = infinity/2 = infinity ...

Thus **N!** is infinity whenever **N** is a negative
integer.
The fact that Stirling's formula for **(z-1)!** has zero radius
of convergence as a series in **1/z** is due to these
infinities. A non-zero radius
of convergence in **1/z** would have to include some
negative values near zero - hence some large negative integers.
Since this factorial function **(z-1)!** (also known as the
Gamma function) has infinities at all the negative integers, it can't
be described by a convergent series there.
It can't converge for 10! because it mustn't converge for (-12)!

- Elastic Theory has Zero Radius of Convergence.
- What is the radius of convergence?
- Famous Asymptotic Series

Last modified: May 24, 1997

James P. Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).