Discrete Mathematics Demystified

CHAPTER 12
Sequences
12.1 Introductory Remarks
Many physical and mathematical quantities are best understood by using approximations.
Sometimes approximating values are given as limits. For instance, suppose
that we want to paint the (infinitely many) squares shown in Fig. 12.1. Suppose
also that 1 gallon of paint covers 500 square feet. How much paint will we need?
We see that the first square has interior area 900 square feet. The first two squares
have area 900 + 90 = 990 square feet. The first three squares have area 900 + 90 +
9 = 999 square feet. The first four squares have area 900 + 90 + 9 + 0.9 = 999.9
square feet. The pattern is clear. We have a list of approximations, each given by a
sum of finitely many numbers. The approximations seem to tend to 1000 . Therefore
it seems that we will need enough paint to cover 1000 square feet, or 2 gallons of
paint ([we are of course conveniently ignoring the fact that the (100 100 )th square
will be too small to hold even one molecule of paint)].
The purpose of this chapter is to turn the above intuitive discussion into careful
mathematics. A list of approximating values will be called a “sequence.” We want
especially to study the situation when the sequence consists of “partial sums” (as in