1. 8.1: Sequences Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008 Craters of the Moon National Park, Idaho

2. A sequence is a list of numbers written in an explicit order. n th term 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 100 th term, plug 100 in for n :

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

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

6. An geometric sequence has a common ratio between terms. Geometric sequences can be defined recursively: Example: 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? The sequence converges and its limit is 2.

9. Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero.  Don’t forget to change back to function mode when you are done plotting sequences.