9.1 Sequences

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.

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 :

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.

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

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

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.

Sequence Graphing on the Ti-89 Change the graphing mode to sequence: MODE Graph…….4 ENTER

Example: Plot Y= Use the key to enter the letter n. alpha Leave ui1 blank for explicitly defined functions.

WINDOW

GRAPH

The previous example was explicitly defined. Now we will use a recursive definition to plot the Fibonacci sequence. Y= Use the key to enter the letters u and n. alpha Enter the initial values separated by a comma (even though the comma doesnt show on the screen!)

WINDOW

GRAPH You can use F3 Trace to investigate values.

TBLSET TABLE We can also look at the results in a table. Scroll down to see more values.

TABLE Scroll down to see more values.

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.

Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero. Dont forget to change back to function mode when you are done plotting sequences.