1. Copyright © 2011 Pearson Education, Inc. Slide 11.1-1 11.1 Sequences A sequence is a function that has a set of natural numbers (positive integers) as its domain.

2. Copyright © 2011 Pearson Education, Inc. Slide 11.1-2 11.1 Sequences A sequence is often specified by giving a formula for the general term or nth term, a n. Example Find the first four terms for the sequence

3. Copyright © 2011 Pearson Education, Inc. Slide 11.1-3 11.1 Sequences A finite sequence has domain the finite set {1, 2, 3, …, n} for some natural number n. Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 An infinite sequence has domain {1, 2, 3, …}, the set of all natural numbers. Example 1, 2, 4, 8, 16, 32, …

4. Copyright © 2011 Pearson Education, Inc. Slide 11.1-4 11.1 Convergent and Divergent Sequences A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number. A sequence that is not convergent is said to be divergent.

5. Copyright © 2011 Pearson Education, Inc. Slide 11.1-5 11.1 Sequences and Recursion Formulas A recursion formula or recursive definition defines a sequence by –Specifying the first few terms of the sequence –Using a formula to specify subsequent terms in terms of preceding terms.

6. Copyright © 2011 Pearson Education, Inc. Slide 11.1-6 11.1 Series and Summation Notation is the sum of the first n terms of the sequence. = a 1 + a 2 + a 3 + … + a n

7. Copyright © 2011 Pearson Education, Inc. Slide 11.1-7 Summation Properties (a) (b) (c)

8. Copyright © 2011 Pearson Education, Inc. Slide 11.1-8 11.1 Series and Summation Notation A finite series is the sum of the first n terms of the sequence: Infinite series is the sum of all the terms of the infinite sequence:.