http://www.ck12.org Chapter 8. Infinite Series
8.1 Sequences
Learning Objectives
- Demonstrate an understanding of sequences and their terms
- Determine if the limit of a sequence exists and, if it exists, find the limit
- Apply rules, theorems, and Picard’s method to compute the limits of sequences
Sequences (rules, terms, indices)
The alphabet, the names in a phone book, the numbered instructions of a model airplane kit, and the schedule in the
local television guide are examples of sequences people may use. These examples are all sets of ordered items. In
mathematics, a sequence is a list of numbers. You can make finite sequences, such as 2, 4 , 6 ,8. These sequences
end. You can also make infinite sequences, such as 3, 5 , 7 , 9 ,...,which do not end but continue on as indicated by
the three dots. In this chapter the wordsequencerefers to an infinite sequence.
Each term in a sequence is defined by its place of order in the list. Consider the sequence 3, 5 , 7 , 9 ,...The first term
is 3 because it belongs to place 1 of the sequence. The second term is 5 because it belongs to the second place of
the sequence. Likewise, The third term is 7 because it is in the third place. Notice that there is a natural relationship
between the counting numbers, or the positive integers, and the terms of the sequence. This leads us to the definition
of asequence.
Sequence
Asequenceis a function from the domain of the set of counting numbers, or positive integers, to the range which
consists of the members of a sequence.
A sequence can be denoted by{an}or bya 1 ,a 2 ,a 3 ,a 4 ,...,an,...
The numbersa 1 ,a 2 ,a 3 ,a 4 ,...,an,...that belong to a sequence are calledtermsof the sequence. Each subscript of
1 , 2 , 3 ,...on the termsa 1 ,a 2 ,a 3 ,a 4 ,...refers to the place of the terms in the sequence, or theindex.The subscripts
are called theindicesof the terms. We assume thatn= 1 , 2 , 3 ,...,unless otherwise noted.
Instead of listing the elements of a sequence, we can define a sequence by arule,or formula, in terms of the indices.
Example 1
The formulaan=^1 nis a rule for a sequence.
We can generate the terms for this rule as follows:
n 1 2 3 4 ...
an=^1 n^11 = 1 12 13 14 ...Example 2
Consider the sequence rulean=nn+^21.