The Art and Craft of Problem Solving

9.4 POWER SERIES AND EULERIAN MATHEMATICS 349

appear exactly once, because prime factorization is unique (see the discussion of the

fundamental theorem of arithmetic in Section 7.1). Thus

1 1 1 1

1 + 2" + "3 + 4" + :5 + ... = S 2 S 3 S 5 S 7 S I I ....

For each k, we have Sk = 1/( I - 1/ k) = k/ (k -I), which is finite. But the harmonic

series is infinite. So it cannot be a product of finitely many Sk. We conclude that there
are infinitely many primes! _

Our penultimate example is also due to Euler. Here the tables are turned: ideas
from polynomial algebra are inappropriately applied to a calculus problem, resulting in
a wonderful and correct evaluation of an infinite series (although in this case, complete
rigorization is much more complicated). Recall that the zeta function (see page 16 2)
is defined by the infinite series

I I 1

�(s):= TS+ 2 s + 3 s + ....

Example 9.4.8 Is there a simple expression for �(2) = 1 + ; 2 +
3

1

2 + ...?

Solution: Euler's wonderful, crazy idea was inspired by the relationship between
zeros and coefficients (see Section 5.4), which says that the sum of the zeros of the
monic polynomial

;(I + an_I;(I-l ;(I-I + ... + alx + ao

is equal to -an -I; this follows from an easy argument that examines the factorization
of the polynomial into terms ofthe form (x - n), where each ri is a zero.

Why not try this with functions that have infinitely many zeros? A natural can­

didate to start with is sinx, because its zeros are x = k1r for all integers k. But we

are focusing on squares, so let us modify our candidate to sin y'X. The zeros of this
function are 0, 1r^2 , 41r^2 , 91r^2 , .... Since we will ultimately take reciprocals, we need to
remove the 0 from this list. This leads to our final candidate, the function

f(

).=

siny'X

x.

y'X

'

which has the zeros 1r^2 ,41r^2 , 91r^2 , .... Using the methods of Example 9.4.4, we easily
discover that

f x x^2 x^3

(x) = 1-

3!

+

5!

• 7!

+ .... (10)

Since we know all the zeros of f(x), we can pretend that the factor theorem for poly­
nomials applies, and write f(x) as an infinite product of terms of the form x -n^2 1r^2.
But we need to be careful. The product