Lesson 11 1 Sequences Vocabulary Sequence a list

Lesson 11 -1 Sequences

Vocabulary • Sequence – a list of numbers written in a definite order { a 1, a 2, a 3, an-1, an} • Fibonacci sequence – a recursively defined sequence where 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 Find the formula for the general term an of the sequence is ½, ¼, 1/8, 1/16, …. . an = 1 / 2ⁿ

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

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 n→∞ = 1 Therefore the sequence converges The sequence is increasing, since an < an+1 The sequence is bounded above by 1

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 / (3 n – 1) n+1 Lim an = Lim -----n→∞ 3 n - 1 = 1/3 Therefore the sequence converges The sequence is decreasing, since an > an+1 The sequence is bounded below by 1/3

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 2 n 2 Lim an = Lim -----= Lim ------n n e e en n→∞ = 0 Therefore the sequence converges The sequence is monotonic, since it is neither increasing nor decreasing for all n The sequence is bounded below by 0

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