5.1 Sequences

In this section, we introduce sequences and define what it means for a sequence to converge or diverge. We show how to find limits of sequences that converge, often by using the properties of limits for functions discussed earlier. We close this section with the Monotone Convergence Theorem, a tool we can use to prove that certain types of sequences converge.

Terminology of sequences

To work with this new topic, we need some new terms and definitions. First, an infinite sequence is an ordered list of numbers of the form

Each of the numbers in the sequence is called a term. The symbol $n$ is called the index variable for the sequence. We use the notation

to denote this sequence. A similar notation is used for sets, but a sequence is an ordered list, whereas a set is not ordered. Because a particular number $a_n$ exists for each positive integer $n,$ we can also define a sequence as a function whose domain is the set of positive integers.

Let’s consider the infinite, ordered list

This is a sequence in which the first, second, and third terms are given by $a_1=2,$ $a_2=4,$ and $a_3=8.$ You can probably see that the terms in this sequence have the following pattern:

Assuming this pattern continues, we can write the $n\text{th}$ term in the sequence by the explicit formula $a_n=2^n.$ Using this notation, we can write this sequence as

Alternatively, we can describe this sequence in a different way. Since each term is twice the previous term, this sequence can be defined recursively by expressing the $n\text{th}$ term $a_n$ in terms of the previous term $a_{n-1}.$ In particular, we can define this sequence as the sequence $\left\{a_n\right\}$ where $a_1=2$ and for all $n\ge 2,$ each term $a_n$ is defined by the recurrence relation     $a_n=2a_{n-1}.$

Note that the index does not have to start at $n=1$ but could start with other integers. For example, a sequence given by the explicit formula $a_n=f(n)$ could start at $n=0,$ in which case the sequence would be

Similarly, for a sequence defined by a recurrence relation, the term $a_0$ may be given explicitly, and the terms $a_n$ for $n\ge 1$ may be defined in terms of $a_{n-1}.$ Since a sequence $\left\{a_n\right\}$ has exactly one value for each positive integer $n,$ it can be described as a function whose domain is the set of positive integers. As a result, it makes sense to discuss the graph of a sequence. The graph of a sequence $\left\{a_n\right\}$ consists of all points $(n,a_n)$ for all positive integers $n.$ [link] shows the graph of $\left\{2^n\right\}.$