The Art and Craft of Problem Solving

350 CHAPTER 9 CALCULUS

won't work; the constant term is infinite, which is horribly wrong. The power series

(10) tells us that the constant term must equal 1. The way out of this difficulty is to

write the product as

f{x) = (1-�) (1 -�) (1 -�) ...

�2 4 �2 9 �^2 '

(11)

for now the constant term is 1, and when each factor 1 - x / (n^2 �^2 ) is set equal to zero,

we get x = n^2 �^2 , just what we want.

Now it is a simple matter of comparing coefficients. It is easy to see that the

coefficient of the x-term in the infinite product (11) is

_ (�+1+1+ ... ).

�2 4 �2 9 �2

But the corresponding coefficient in the power series (10) is -1/3!. Equating the two,

we have

and thus

-+-+-+^1 1 1 ...^1

=-

�^2 4 �^2 9 �^2 6 '

1 1 �^2

�(2) = 1 +

22

+

32

+ ... =

6·

Beauty, Simplicity, and Symmetry: The Quest for a Moving Curtain

Our final example is an intriguing probability question that we used in a math con­

test. We are indebted to Doug Jungreis for bringing it to our attention. This problem

has several solutions. One, using calculus and generating functions, is startling in its

beauty. Yet another, more prosaic argument, sheds even more light on the problem.

Example 9.4.9 (Bay Area Math Meet 20oo) Consider the following experiment:

• First a random number p between 0 and 1 is chosen by spinning an arrow around
  a dial that is marked from 0 to 1. (This way, the random number is "uniformly
  distributed"-the chance that p lies in the interval, say, from 0.45 to 0.46 is
  exactly 1/100; and the chance that p lies in the interval from 0.32 4 to 0.335 is
  exactly 11/1000, etc.)


• Then an unfair coin is built so that it lands "heads up" with probability p.


• This coin is then flipped 2000 times, and the number of heads seen is recorded.
  What is the probability that exactly 1000 heads were recorded?
  Solution 1: Generating Functions. Suppose there are n tosses. If the value p were

fixed, basic probability theory would tell us that the probability of seeing k heads in n

tosses is just