MiraCosta College



Math 155, Lecture Notes- Bonds Name____________

Section 9.1 Sequences

A sequence is a function whose domain is the set of positive integers. It will usually be denoted with subscript notation rather than function notation. You can use your graphing calculator in “sequence mode” to plot terms and create tables that show terms in a sequence.

[pic] [pic]

An entire sequence can be denoted as [pic] .

Ex. 1:

[pic]

Ex. 2:

[pic]

Some sequences are recursively defined.

Ex. 3:

[pic] is defined as [pic] and [pic].

[pic]

For the majority of the chapter, we’ll be looking at sequences that have limiting values. These sequences are said to converge.

Ex. 4:

[pic] This sequence converges to 0.

[pic]

If we plot the terms of a convergent sequence, we will see a “horizontal asymptote.” That is, we will see the sequence exhibit asymptotic behavior.

[pic]

Ex. 5:

Given: [pic]

Consider [pic]

This sequence converges to 1.

[pic]

In other words, if a sequnce [pic] ”agrees” with a function [pic] at every positive integer, and if [pic] as [pic], then [pic] as well.

Ex. 6:

Given:[pic] Consider [pic]

[pic]

New Notation: Factorial !

Try working with these on your graphing calculator.

[pic]

[pic]

Ex. 7:

Given: [pic] Consider [pic]

[pic]

Ex. 8:

Given: [pic] Consider [pic]

Ex. 9:

Given: [pic] Consider [pic]

Ex. 10:

Given: [pic] Consider [pic]

Ex. 11:

Given: [pic] Consider [pic]

Ex. 12:

Given: [pic] Consider [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Ex. 13:

Given: [pic] Consider [pic]

[pic]

From the graph of [pic], for [pic], we can see that the function is bounded above by [pic] and bounded below by [pic]. Therefore, by Theorem 9.5, [pic] is a convergent sequence, since [pic] is bounded and monotonic for [pic].

Ex. 14: The Fibonacci Sequence

Consider the sequence is defined by [pic] with [pic] and [pic].

[pic]

This is the Fibonacci Sequence.

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