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Triple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Augustus De Morgan (1806-1871) was a significant Victorian Mathematician who made
contributions to Mathematics History, Mathematical Recreations, Mathematical Logic, Calculus,
and Probability and Statistics. He was an inspiring Mathematics professor who influenced many
of his students to join the profession. There have been a number of articles supporting his choice
as the best Mathematics professor of the 19th Century (A. Rice, What makes a great mathematics
teacher? The Case of Augustus De Morgan, Amer. Math. Monthly 106 (1999), pp. 534-552).
One of De Morgan’s significant books was his Essay on Probabilities and on their Application
to Life Contingencies and Insurance Offices (Longman, Brown, and Green, London, 1838). The
intent of the book was to make Probability Theory accessible to students who no mathematical
training beyond arithmetic. Thus, while De Morgan gives a logical presentation, he avoids
higher level mathematics, and sticks to an algorithmic, computational approach. Because
factorials of whole numbers play an important role in Probability Theory, and were difficult to
compute in the 19th Century for large whole numbers, De Morgan introduces the following
algorithm (pp. 15-16) to approximate n!, where n is a whole number. Note that [n] in De
Morgan’s notation means n factorial.
I wondered why his algorithm for n! worked. That led me to an Intermediate
Algebra/College Algebra investigation of the algorithm, which led to Stirling’s Formula,
which approximates n!. Having recently taught Calculus II and Calculus III at my
college, I wondered why Stirling’s Formula so closely approximated n!. That in turn led
to the Gamma Function (significantly developed by Euler), a proof by Mathematical
Induction (De Morgan introduced the term Mathematical Induction, and made the process
rigorous) of the Gamma Function for n!, and a proof of Stirling’s Formula using double
integrals in Polar Coordinates, and Maclaurin Series. In the process, I also learned some
Mathematics History and discovered some interesting exercises for Intermediate Algebra,
College Algebra, Calculus II, and Calculus III.
From De Morgan to Stirling
Stirling’s approximation of n! was discovered by James Stirling (1692-1770), a Scottish
Mathematician. Stirling’s approximation states that
n
n ! ~ 2n  
e
n
Refering to De Morgan’s algorithm for n!, and noting that .4342945 is an approximation of
log e, and .7981799 is an approximation of log 2 , we have:
n
(1) Take the log of the number, and subtract .4342945: log n – log e = log  
e
n
n
n
(2) Multiply by n: n log   = log  
e
e
(3) Take log n and add .7981799 = log n + log 2 = log 2n 
1
(4) Take half of the latter result: log 2n  log 2n
2
n
n
(5) Add the results of (2) and (4): log   + log 2n = log
e
n
2n  
e
n
n
n
(6) But this an approximation of the log n!, so n ! ~ 2n   , Stirling’s Formula.
e
This would be a nice exercise for Intermediate Algebra or College Algebra students. Next, I
wondered why Stirling’s Formula approximated n!
From Stirling to Euler and the Gamma Function
Because the Gamma Function gives n! for whole numbers, I next looked at the Gamma Function;
Euler significantly contributed to its development. The Gamma Function provides a formula for
the factorial of any complex number with a positive real part. In my case, I was interested in the
factorial of whole numbers. So, I sought an inductive proof of the Gamma Function for whole
numbers, a proof that could be used in Calculus II. The Gamma Function states for whole
numbers n:

n !   x n e  x dx
0
To inductively prove that it is true for whole numbers:
(1) Verify it is true for n=0. 0! =1 and

(2) Assume k! =

0


0

x0e x dx =  e x dx =1
0
x k e x dx

(3) Then (k+1)! should equal  x k 1e x dx . Using Integration by Parts with
0
u = x k 1 and dv = e  x dx we get

 x k 1e x |
0

0
x k 1e x dx =

+ (k  1)  xk e x dx  0  (k 1) k !  ( k 1)!
0

where

 x k 1e x |
= 0 by L’Hospital’s Rule.
0
So, having faith in the Gamma Function as accurately producting n! for whole numbers, I wanted
to prove that Stirling’s formula was an approximation of the Gamma Function, and thus an
approximation of n!. That led me to an intermediate result using Calculus III and polar
substitution that I needed before I established that Stirling’s formula approximated the Gamma
Function.
An Intermediate Result
Before I could go use the Maclaurin Series on the Gamma Function, I had to establish an
intermediate result:

e
 x2
2n
dx = 2n

Because f(x) = e

e
Let I =
 x2
2n


Then A =  e
2

A =
e
 x2
2n
0
is an positive even function, where n is a whole number,

x
1
dx and A= I =  e 2 n dx .
2
0
 x2
2n
0
2
 x2
2n

dx  e
 x2
2n
2
dx which by Fubini’s Theorem can be rewritten
0

dx  e
 y2
2n

dy =
0
e
 x2  y 2 


 2n 
dxdy .
0 0
Using polar substitution ( x 2  y 2  r 2 and dxdy  rdrd ) we get

2 
2
A =
e
 r2 
 
 2n 
rdrd , where the limit on  is 0 to

2
because we are in Quadrant 1.
0 0
Evaluating this double integral with u substitution ( u 
Since I = 2A =
4

2
r 2
), we get A2 =
2n

2
n or A =

2
n.
n = 2n . This would be a good exercise for Calculus III students.
From the Gamma Function (Euler) to Maclaurin (which leads us back to Stirling)
My goal was now to show that Stirling’s Formula is an approximation of n!, and therefore an
approximation of the Gamma Function. To establish this, I did the following:

(1) n! =  x n e  x dx by the Gamma Function where n is a whole number.
0
(2) Let u = (x-n) or x= (u+n). Also du = dx. Substituting, we get
n

n
  u    u n
u 
n
(3)   n   1  e   du   n n   1 e u e  n du   
n
n 
e
 n 
n

n 
n
 u  u
 n  n  1 e du
n
u 
u 
(4) Look at ln   1  = n ln   1 .
n 
n 
2
u  u u 
(5) The first two terms of the Maclaurin Series for ln   1 =   2  and
 n   n 2n 
 u u2  
u2 
(6) n   2  =  u  
2n 
 n 2n  
(7) Substituting the Maclaurin Series approximation back into (3) we get (as n   )
n
(8) n! ~  
e
n 
e
n
 u2 
 u  
 2 n  u
n
e du   
e
n 
e
 u2 
  
 2n 
n
du = by the Intermediate Result

which takes us back to Stirling’s Formula.
This would be a good exercise for Calculus II students.
n
2n   ,
e
A final investigation
As n   , how closely does Stirling’s Formula approximate n!. One way to look at this is to
n
n
2n  
 e  . Because of n!, it would be impossible to apply L’Hospital’s Rule.
look at limn
n!
The reader is invited to tackle the problem on their own. However, using Scientific Notebook,
n
limn
n
2n  
 e  = 1, indicating that as n increases, Stirling’s Formula becomes n!.
n!
Conclusion
I was interested in reading De Morgan’s work on probability (a first edition I just added to my
antiquarian math book collection) to get an historical perspective on the subject. Who would
imagine that in so doing, I would end up developing enrichment exercises for classes from
Intermediate Algebra through Calculus III, and learning more Mathematics History along the
way.