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. 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

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

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

11-1 Example 3 sequence n th term is n/n+1 n Lim a n = Lim n + 1 = 1 Therefore the sequence converges n→∞ 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. The sequence is bounded above by 1 The sequence is increasing, since a n < a n+1

11-1 Example 4 sequence n th term is n + 1 / (3n – 1) n + 1 Lim a n = Lim n - 1 = 1/3 Therefore the sequence converges n→∞ 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. The sequence is bounded below by 1/3 The sequence is decreasing, since a n > a n+1

11-1 Example 5 sequence n th term is n² e -n n² 2n 2 Lim a n = Lim = Lim = Lim e n e n e n = 0 Therefore the sequence converges n→∞ 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. The sequence is bounded below by 0 The sequence is monotonic, since it is neither increasing nor decreasing for all n

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