Lesson 11-1

Sequences

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

11-1 Example 1

sequence is ½, ¼, 1/8, 1/16, …..

an = 1 / 2ⁿ

Find the formula for the general term an of the sequence

11-1 Example 2

sequence is ½, ¼, 1/6, 1/8, …..

an = 1 / 2n

Find the formula for the general term an of the sequence

11-1 Example 3

sequence nth term is n/n+1

nLim an = Lim -------- n + 1

= 1

Therefore the sequence converges

n→∞ 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 an < an+1

11-1 Example 4

sequence nth term is n + 1 / (3n – 1)

n + 1Lim an = Lim ---------- 3n - 1

= 1/3

Therefore the sequence converges

n→∞ 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 an > an+1

11-1 Example 5

sequence nth term is n² e-n

n² 2n 2Lim an = Lim ------ = Lim ------ = Lim ------- en en en

= 0

Therefore the sequence converges

n→∞ 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