Sequences Lesson 8.1

Definition A succession of numbers Example Listed according to a given prescription or rule Typically written as a1, a2, … an Often shortened to { an } Example 1, 3, 5, 7, 9, … A sequence of odd numbers

Finding the nth Term We often give an expression of the general term That is used to find a specific term What is the 5th term of the above sequence?

Sequence As A Function Define { an } as a function Domain set of nonnegative integers Range subset of the real numbers Values a1, a2, … called terms of the sequence Nth term an called the general term Some sequences have limits Consider

Converging Sequences Note Theorem 9.2 on limits of sequences Limit of the sum = sum of limits, etc. Finding limit of convergent sequence Use table of values Use graph Use knowledge of rational functions Use L'Hopital's Rule

Divergent Sequences Some sequences oscillate Others just grow beyond bound

Determining Convergence Manipulate algebraically Simplify and take the limit conjugate expressions

Determining Convergence Consider Use l'Hôpital's rule to take the limit of the function Note we are relating limit of a sequence from the limit of a continuous function

Bounded, Monotonic Sequences Note difference between Increasing (decreasing) sequence Strictly increasing (decreasing) sequence Note concept of bounded sequence Above Below Bounded implies convergent Both

Assignment Lesson 9.1 Page 602 Exercises 1 – 93 EOO