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  • Find the formula for the general term of a sequence.
  • Calculate the limit of a sequence if it exists.
  • Determine the convergence or divergence of a given sequence.

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

a 1 , a 2 , a 3 ,… , a n ,… .

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

{ a n } n = 1 , or simply { a n } ,

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

2 , 4 , 8 , 16 , 32 ,… .

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:

a 1 = 2 1 , a 2 = 2 2 , a 3 = 2 3 , a 4 = 2 4 , and a 5 = 2 5 .

Assuming this pattern continues, we can write the n th term in the sequence by the explicit formula a n = 2 n . Using this notation, we can write this sequence as

{ 2 n } n = 1 or { 2 n } .

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 th term a n in terms of the previous term a n 1 . In particular, we can define this sequence as the sequence { a n } where a 1 = 2 and for all n 2 , each term a n is defined by the recurrence relation     a n = 2 a n 1 .

Definition

An infinite sequence { a n } is an ordered list of numbers of the form

a 1 , a 2 ,… , a n ,… .

The subscript n is called the index variable    of the sequence. Each number a n is a term    of the sequence. Sometimes sequences are defined by explicit formulas , in which case a n = f ( n ) for some function f ( n ) defined over the positive integers. In other cases, sequences are defined by using a recurrence relation    . In a recurrence relation, one term (or more) of the sequence is given explicitly, and subsequent terms are defined in terms of earlier terms in the sequence.

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

a 0 , a 1 , a 2 ,… .

Similarly, for a sequence defined by a recurrence relation, the term a 0 may be given explicitly, and the terms a n for n 1 may be defined in terms of a n 1 . Since a sequence { a n } 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 { a n } consists of all points ( n , a n ) for all positive integers n . [link] shows the graph of { 2 n } .

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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