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Lesson 11-1

Lesson 11-1. Sequences. Vocabulary. Sequence – a list of numbers written in a definite order { a 1 , a 2 , a 3 , a n-1 , a n } Fibonacci sequence – a recursively defined sequence where the third term is defined by the sum of the preceding two terms and so on.

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Lesson 11-1

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  1. Lesson 11-1 Sequences

  2. Vocabulary • Sequence – a list of numbers written in a definite order { a1, a2, a3, an-1, an} • Fibonacci sequence – a recursively defined sequence where the third term is defined by the sum of the preceding two terms and so on. • Sequence converges – if it limit exists as n approaches infinity • Sequence diverges – if it limit does not exist as n approaches infinity • Increasing – if an < an+1 for all n ≥ 1 • Decreasing – if an > an+1 for all n ≥ 1 • Monotonic – neither increasing nor decreasing • Bounded Above – if M ≥ an for all n ≥ 1 • Bound Below – if m ≤ an for all n ≥ 1

  3. 11-1 Example 1 Find the formula for the general term an of the sequence sequence is ½, ¼, 1/8, 1/16, ….. an = 1 / 2ⁿ

  4. 11-1 Example 2 Find the formula for the general term an of the sequence sequence is ½, ¼, 1/6, 1/8, ….. an = 1 / 2n

  5. 11-1 Example 3 Examine the sequence below and determine if it converges or diverges, if it is increasing, decreasing, or monotonic and if it is bounded above or below. sequence nth term is n/n+1 n Lim an = Lim -------- n + 1 = 1 Therefore the sequence converges n→∞ n→∞ The sequence is increasing, since an < an+1 The sequence is bounded above by 1

  6. 11-1 Example 4 Examine the sequence below and determine if it converges or diverges, if it is increasing, decreasing, or monotonic and if it is bounded above or below. sequence nth term is n + 1 / (3n – 1) n + 1 Lim an = Lim ---------- 3n - 1 = 1/3 Therefore the sequence converges n→∞ n→∞ The sequence is decreasing, since an > an+1 The sequence is bounded below by 1/3

  7. 11-1 Example 5 Examine the sequence below and determine if it converges or diverges, if it is increasing, decreasing, or monotonic and if it is bounded above or below. sequence nth term is n² e-n n² 2n 2 Lim an = Lim ------ = Lim ------ = Lim ------- en en en = 0 Therefore the sequence converges n→∞ n→∞ The sequence is monotonic, since it is neither increasing nor decreasing for all n The sequence is bounded below by 0

  8. Homework Pg 710 – 712: problems 4, 11, 16, 21

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