Why Read This Book? - Index of

Sequences of Real Numbers
6.1 Sequences Defined
6
We can approach the subject of sequences in several ways. Let’s move from an
informal to a more formal way. A sequence of real numbers is an ordered list
〈a1,a2,a3,...,an,...〉 (6.1)
Listing the terms of a sequence in order might bring back memories of showing a
set A is countable, for this listing is equivalent to constructing a function f : N onto
−−→
A, where f(1) = a1, f(2) = a2, and so on. A sequence can therefore be defined as
a function from the positive integers into the reals. The notation 〈an〉 ∞ n=1 is one
standard way of denoting a sequence. Beginning the list with a1 is a matter of
convenience. There might be times when it seems more natural to begin with a0.
If there is a formula for an, say, for example, an = 1/n, we have
〈 1 n 〉∞ n=1 =〈1, 1 2 , 1 3 ,...〉 (6.2)
We must distinguish (6.2) from {1/n} ∞ n=1 , which is merely the set of elements of
the sequence 〈1/n〉 and does not have the ordering of the elements as one of
its defining characteristics. Saying an = 1/n makes the idea of a sequence as a
function more natural, for we are talking about nothing other than f : N → R
defined by f(n) = 1/n, and the set {1/n} can then be thought of as the range of
the sequence. Then, if we want to visualize the sequence graphically, we can, as in
Figure 6.1. Such a graph will help in visualizing limits in Section 6.2. Example 6.1.1
presents some examples of sequences we will refer to later.
Example 6.1.1
1. The sequence 〈an〉 ∞ n=1 defined by an = 1 2 + (−1)n · 1 2 is the sequence
〈0, 1, 0, 1, 0, 1,...〉.
2. Letting an = n + (−1) n for n ≥ 0 generates 〈1, 0, 3, 2, 5, 4, 7, 6,...〉.
3. If an = sin nπ, then we generate the sequence 〈0〉.
185