1 Sequences in R

Our discussion begins with a very basic concept in mathematics: sequences. Probably most people have at least a general intuitive idea of what a sequence is. One simple example of a sequence is

while another more irregular sequence is

In the second case, it is not possible to predict what the next entry will be after -6, but nonetheless, both examples seem to satisfy the essential sense of what a sequence is. What is needed now is an exact mathematical definition:

Definition.

A sequence is a function a:AR where either A=Z+:={1,2,3,.} or A=N:={0,1,2,3,.} (the natural numbers) or perhaps A is any countable set.1 Thus, either

or

where in both cases an?R. We refer to an as the n-th element or entry of the sequence.

Remarks.

1.

Notice that a sequence differs from a set in that a sequence has an order, whereas a set is simply a collection of elements. Technically, there is no first element in a set, but there is in a sequence.

2.

This is not the most general definition of the word "sequence," but it is easily general enough for our purposes. It contains all the key ideas to discuss sequences. In particular, we will not discuss finite sequences.

3.

Notice that for our purposes, sequences begin either with n=0 or with n=1. Beginning with a zeroth element is less common, but it occurs sometimes when a zeroth element makes sense. Among other places, this happens in computer science. Still, unless there is some specific reason to do otherwise, let A=Z+.

A sequence can be depicted graphically on the real line as in Figure 1.1.

Figure 1.1 A sequence on the real line. Here several elements of the sequence are shown in blue. Notice that there is no particular pattern to this sequence. Which of the elements are positive, and which are negative?

1.1 Convergent sequences

Now that we have a definition for sequences, we can introduce the central concept of convergence, starting with a basic example.

Example 1.1.1.

Suppose that an:=1/n (i.?e., an is defined equal to 1/n), so that

Notice that this sequence must start with n=1 since a0 is undefinable using the given formula. Also notice that an decreases toward zero as n increases toward infinity, even though an?0 for any n?Z+. In symbols, that is an?0 as n?8.

All of this may seem clear intuitively, but as we will see, intuition is not always so clear. So again an exact mathematical definition is needed.

Definition.

A sequence {an}?R converges to a limit L?R iff given e>0, ?N?Z+ such that |an-L|<e whenever n>N. Written with a minimal amount of symbols, this definition is "a sequence {an} of real numbers converges to a limit L in the reals if and only if given e>0, there exists N in the positive integers such that |an-L|<e whenever n>N." The sequence {an} diverges iff it does not converge.

Notation.

The notation to indicate that a sequence {an} converges to a limit L is

or more briefly

Remark.

In this definition, e can be any positive real number, but it is generally thought of as a small positive real number. The definition works because e can be an arbitrary small positive real number. A schematic diagram of the relationship between e, L and the elements of the sequence ak is shown in Figure 1.2.

Figure 1.2 A sequence approaching a limit L on the real line. The early elements of the sequence through aN can be outside the interval (L-e,L+e), but starting with aN+1, all of the elements must lie inside the interval. In particular, when n>N, then an must lie in the interval. Where might aN+5 be on this number line? If e is made smaller, then likely N will have to be made larger to keep the elements of the sequence from aN+1 onward in the new smaller interval.

Example 1.1.1.

Returning to our first example, suppose again that an:=1/n. Does the sequence {an} converge, and if so, to where? For a given e, what value of N guarantees that the convergence definition is satisfied?

Answer. Notice that there are two questions here: The first is a calculus question (Where does the sequence converge to? What is the value of L?), while the second is an analysis question (How does one show that the definition is satisfied?). In this case, the calculus question is relatively easy, and in fact we have already mentioned the answer: L=0. The second question is more interesting: Given e>0, what value of N is needed to guarantee that |an-L|<e when n>N? Notice that here

This may seem obvious, but notice that in fact, it gives us the answer we are looking for: define N:=?1/e?, where ?x? is the ceiling function or next greater integer function for x (the least integer greater than or equal to x). It is important that this ceiling function be used because N must be an integer, but for most values of e, its reciprocal 1/e is not itself an integer. Then for this example, |an-L|<e whenever n>N:=?1/e?=1/e.

In summary, the limit here is L=0 and setting N:=?1/e? allows the definition of convergence to be satisfied.

Remarks.

1.

Notice that the value of N increases as e decreases; this is almost always the case. Expect e to be in the denominator in the expression for N.

2.

Notice the direction of the double arrow above. Although we started on the left working on |an-L|, we need to keep in mind that this is the conclusion to be reached provided that n is large enough. So a backward implication (a backward double arrow) is what is needed. Indeed, the symbol "?" is perhaps best read as "whenever". Thus, the implication above becomes |an-L|<e whenever n>1/e.

The first example above is relatively straightforward; the next two examples are more interesting ("more interesting" means "harder")....