A PRIMER OF ANALYTIC NUMBER THEORY: From Pythagoras to ...

I2.4 Geometric Series 125
can be detected by considering the residue of f ′ (z)/f (z). When residues can
be computed by means of integrals, this provides a way of counting the total
number of zeros of f (z) in any given region.
Exercise I2.3.1. Compute the Laurent expansion of 2/(x 2 − 1) at x =−1.
Do the same for x = 0.
Exercise I2.3.2. What value should you assign to the function x 2 /(cos(x) −
1) at x = 0?
Exercise I2.3.3. Suppose a function f (z) has a zero of order N at z = a,
that is,
f (z) = aN (z − a) N + O((z − a) N+1 ), with aN �= 0.
Compute the residue of f ′ (z)/f (z)atz = a.
I2.4. Geometric Series
The ancient Greeks created a lot of beautiful mathematics, but they were not
very comfortable with the idea of infinity. Zeno’s paradox is an example.
Zeno of Elea (circa 450 b.c.) is known for the following story about the
Tortoise and Achilles. (Achilles was the hero of Homer’s Iliad.) Achilles and
the Tortoise were to have a race, and Achilles gave the Tortoise a head start
of one kilometer. The Tortoise argued that Achilles could never catch up,
because in the time it took Achilles to cover the first kilometer, the Tortoise
would go some further small distance, say one tenth of a kilometer. And in
the time it took Achilles to cover that distance, the Tortoise would have gone
still further, and so forth. Achilles gets closer and closer but is never in the
same place (asserts Zeno).
Actually, the mathematicians of the ancient world had the techniques
needed to resolve the paradox. Archimedes knew the Geometric series
1 + x + x 2 +···+x n n+1 1 − x
= . (I2.9)
1 − x
Exercise I2.4.1. The formula (I2.9) was derived in Exercise 1.2.10, using
finite calculus. But there is a shorter proof, which you should find. Start by
multiplying
(1 − x)(1 + x + x 2 ···+x n ).