1. 8.1: Sequences

2. nth term A sequence is a list of numbers written in an explicit order.
Any real-valued function with domain a subset of the positive integers is a sequence.
If the domain is finite, then the sequence is a finite sequence.
In calculus, we will mostly be concerned with infinite sequences.

3. A sequence is defined explicitly if there is a formula that allows you to find individual terms independently.
Example:
To find the 100th term, plug 100 in for n:

4. A sequence is defined recursively if there is a formula that relates an to previous terms.
Example:
We find each term by looking at the term or terms before it:
You have to keep going this way until you get the term you need.

5. An arithmetic sequence has a common difference between terms.
Example:
Arithmetic sequences can be defined recursively:
or explicitly:

6. An geometric sequence has a common ratio between terms.
Example:
Geometric sequences can be defined recursively:
or explicitly:

7. Example:
If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term.

8. You can determine if a sequence converges by finding the limit as n approaches infinity.
Does converge?
(L’Hopital)
The sequence converges and its limit is 2.

9. If the sequence has a limit as n approaches infinity, it converges.
If the sequence does not have a limit, we say it diverges.

10. Absolute Value Theorem for Sequences
If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero.
Sandwich Theorem for Sequences
If and there is an integer N for which an ≤ bn ≤ cn for all n > N, then

11. The calculator can graph sequences defined recursively or explicitly.
Graph a sequence defined explicitly by an = (1/2)n
1. Select Seq and Dot modes
2. Enter the expression into y =
3. Press Window and enter the following values:
nMin = 1 PlotStart = 1 xmin = 0 ymin = 0
nMax = 10 PlotStep = 1 xmax = 10 ymax = 1
xscl = 1 yscl = 1
4. Press Graph

12. Graph a sequence defined recursively by
an = (1/2)an–1
a1 = (1/2)
1. Select Seq and Dot modes
Enter the expression into y =
(the condition a1 = (1/2) goes into u(nMin)
3. Press Window and enter the following values:
nMin = 1 PlotStart = 1 xmin = 0 ymin = 0
nMax = 10 PlotStep = 1 xmax = 10 ymax = 1
xscl = 1 yscl = 1
4. Press Graph