Sequences and series: an introduction

You should be familiar with the usual number systems of mathematics, the natural numbers, the integers, the rationals, the reals, and the complex numbers, and be comfortable with using the symbols , , , , , for the set of such numbers in each of these systems. Most of these web pages will be about and you will learn many important properties of real numbers here. However, to describe and work with the reals we will also require familiarity with , and . A very small amount of the material on these pages discusses too. A complex number can simply be regarded as a pair of real numbers (the real and the imaginary part) with specially defined arithmetic operations.

The normal set-theoretical language is very important for any detailed discussion of the reals and the other number systems. We shall use this with the usual notation, including: curly brackets or braces to define sets; intersection, union, and set-difference of two sets and , , , (this last is sometimes written -); elementhood (when an object is an element of a set), written ; containment of sets (when a set is a subset of another), , and proper containment, ; and the empty set, denoted by . I shall never use in these pages but if you want to use it I recommend that you use it to mean the same as since this is the traditional meaning of the symbol.

There are two competing conventions for , whether to include 0 or not. I am firmly in the first camp, and for me =0,1,2,3, . I am aware you may be reading material written by, or studying with, another mathematician who prefers the other convention. Where possible I will write text, examples, exercises, in such a way that it doesn't matter which you are thinking of. In the few occasions when it does matter, I will try to remind you which my is.

You should be familiar with the idea of an interval on the set of reals. These are sets defined in one of the following ways.

In these definitions, the left end point in the round-bracket cases ( in (, and (,)) is allowed to be -, and the the right end point in the round-bracket cases ( in ,) and (,)) is allowed to be +.

One of the most important functions on the real numbers that gets the subject of analysis going is the absolute value function. You should already have seen this, and you must be sure you understand its definition and can use it accurately in inequalities, etc.

Note that this definition is by cases. Therefore all statements using absolute value will be expected to be proved using arguments that split into several different cases. Absolute value will be discussed in detail later when we talk about the very important triangle inequality.

Another important function is the integer-part function. This is written or and is equal to the greatest integer such that . The existence of such an integer follows from the Archimedean Property of the reals (which will be discussed later) that says for every real number there is an integer greater than , and the Minimal Counter-Example form of induction, already discussed, which says there is a least such . Then for this we have -1 (since -1 and is least such that ) and -1 is .

A variation is which is equal to the least integer such that . Integer part functions will be used a lot and you need to be comfortable with them. They are also important in more abstract settings, and we will look at them again in relation to the Archimedean Property of the reals.

Another important and much-used piece of terminology is that of increasing and decreasing functions. Again, we will have quite a lot to say about these later on, but we will need the terminology from the start.

Let be a subset of the reals, such as an interval (,) or the set of natural numbers, integers, rationals, etc. We say that a function is increasing if whenever , and then () () .

There are useful variations of this. Again, let and . We say that is decreasing if whenever , and then () () . We say that is nonincreasing if whenever , and then () () . We say that is nondecreasing if whenever , and then () () .