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Sequences

Sequences. Lesson 8.1. Definition. A succession of numbers Listed according to a given prescription or rule Typically written as a 1 , a 2 , … a n Often shortened to { a n } Example 1, 3, 5, 7, 9, … A sequence of odd numbers. Finding the n th Term.

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Sequences

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  1. Sequences Lesson 8.1

  2. Definition • A succession of numbers • 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

  3. 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?

  4. 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

  5. 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

  6. Divergent Sequences • Some sequences oscillate • Others just grow beyond bound

  7. Determining Convergence • Manipulate algebraically • Simplify and take the limit conjugate expressions

  8. 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

  9. Bounded, Monotonic Sequences • Note difference between • Increasing (decreasing) sequence • Strictly increasing (decreasing) sequence • Table pg 500 • Note concept of bounded sequence • Above • Below Bounded implies convergent • Both

  10. Assignment • Lesson 9.1 • Page 602 • Exercises 1 – 93 EOO

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