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Sect. 9-1 sequences

Sect. 9-1 sequences. An ordered collection of numbers in a prescribed order defined by a function f(n ) The values, a n are called terms. Sequence. Write the first five terms of the sequence a) b ) . Determine the pattern in the sequence

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Sect. 9-1 sequences

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  1. Sect. 9-1 sequences

  2. An ordered collection of numbers in a prescribed order defined by a function f(n) The values, an are called terms Sequence

  3. Write the first five terms of the sequence a) b)

  4. Determine the pattern in the sequence And use pattern to determine the nth term using inductive reasoning Defining Sequences

  5. 2) Write the next two terms for the given sequence A recursively defined sequence: given the first term, all other terms are defined using that term. d1 = 3.5 and {dn}= dn+1

  6. 3) Find next three terms: pattern?

  7. 4) Find the general term an Numerator: start with 1 for first term, add 2 Denominator: powers of 5 Notice terms alternate signs

  8. 5) Write and expression for the nth term

  9. We say that a sequence converges to a limit L if If no limit exists then an diverges. If the terms increase without bound, {an} diverges to infinity Convergence and Divergence of a sequence

  10. Convergence or Divergence?

  11. Convergence and Divergence ?

  12. Properties of Sequences

  13. 6) Evaluate

  14. Squeeze Theorem for Sequences: If {an}, {bn}, and {cn} are sequences and an ≤ bn ≤ cnfor every n and if , then

  15. 7) Determine whether the sequence converges or diverges. If it converges, find the limit. a. b. c.

  16. 8) Determine whether the sequence converges or diverges. If it converges, find the limit. a. b. c.

  17. 9) Determine whether the sequence converges or diverges. If it converges, find the limit

  18. Factorial (!) 10) simplify 11) simplify

  19. Increasing and Decreasing Sequences • A sequence {an} is increasing if • A sequence {an} is decreasing if • A sequence {an} is monotonic if it is either always increasing or always decreasing

  20. Monotonic? Sequences

  21. Bounded Sequences • A sequence {an} is bounded from above if • A sequence {an} is bounded from below if

  22. Bounded Sequences • A sequence {an} is called Bounded • if it is bounded either from above or below. • If a sequence {an} is bounded • and monotonic, • then it converges.

  23. 12) Is the sequence {an} bounded?

  24. Home Work Page 604 # 1,2,3,5, 15-18, 19-22, 25,27,29, 35,39,45, 47, 51, 59, 61, 73, 77, 87, 89 and 91

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