1. Copyright © 2011 Pearson, Inc. 9.4 Day 1 Sequences Goals: Find limits of convergent sequences.

2. Copyright © 2011 Pearson, Inc. Slide 9.4 - 2 What you’ll learn about Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences … and why Infinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

3. Copyright © 2011 Pearson, Inc. Slide 9.4 - 3 Sequence One of the most natural ways to study patterns in mathematics is to look at an ordered progression of numbers, called a sequence. A _____________ sequence has a fixed number of terms. An _________ sequence has an infinite number of terms. Some sequences have a rule that gives the kth number in the sequence, others do not.

4. Copyright © 2011 Pearson, Inc. Example: Finite vs. Infinite Identity the following sequences as finite or infinite. Slide 9.4 - 4

5. Copyright © 2011 Pearson, Inc. Slide 9.4 - 5 A sequence in which the k th term is given as a function of k A sequence defined by giving the first term (or the first few terms) along with a procedure for finding the subsequent terms

6. Copyright © 2011 Pearson, Inc. Slide 9.4 - 6 Example: Defining a Sequence Explicitly

7. Copyright © 2011 Pearson, Inc. Slide 9.4 - 7 Example: Defining a Sequence Recursively Find the first 6 terms and the 100 th term of the sequence defined recursively by the conditions: for all n > 1

8. Copyright © 2011 Pearson, Inc. Your Turn Slide 9.4 - 8

9. Copyright © 2011 Pearson, Inc. Limit of a Sequence Slide 9.4 - 9

10. Copyright © 2011 Pearson, Inc. Slide 9.4 - 10 Example: Finding Limits of Sequences Determine whether each sequence converges or diverges. If it converges, give the limit.

11. Copyright © 2011 Pearson, Inc. Slide 9.4 - 11 Example: Finding Limits of Sequences

12. Copyright © 2011 Pearson, Inc. End Behavior of Sequences degree of numerator = degree of denominator degree of numerator < degree of denominator degree of numerator > degree of denominator Slide 9.4 - 12

13. Copyright © 2011 Pearson, Inc. 9.4 Day 2 & 3 Sequences Goals: Express arithmetic and geometric sequences explicitly and recursively.

14. Copyright © 2011 Pearson, Inc. Slide 9.4 - 14 Sequence Type ArithmeticGeometric Definition Terms Explicit Formula Recursive Relation

15. Copyright © 2011 Pearson, Inc. Slide 9.4 - 15 Example: Arithmetic Sequences

16. Copyright © 2011 Pearson, Inc. Slide 9.4 - 16 Example: Geometric Sequences

17. Copyright © 2011 Pearson, Inc. Slide 9.4 - 17 Example: Constructing Sequences The fifth and ninth terms of an arithmetic sequence are -5 and -17, respectively. Find the first term and a recursive rule for the n th term.

18. Copyright © 2011 Pearson, Inc. Slide 9.4 - 18 Example: Constructing Sequences The third and sixth terms of a geometric sequence are -75 and -9375, respectively. Find the first term, common ratio, and an explicit rule for the n th term.

19. Copyright © 2011 Pearson, Inc. Slide 9.4 - 19 Example: Constructing Sequences The third and sixth terms of a sequence are 48 and 6, respectively. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

20. Copyright © 2011 Pearson, Inc. Slide 9.4 - 20 Example: Constructing Sequences The third and sixth terms of a sequence are 48 and 6, respectively. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

21. Copyright © 2011 Pearson, Inc. Slide 9.4 - 21 Example: Constructing Sequences The third and sixth terms of a sequence are 48 and 6, respectively. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

22. Copyright © 2011 Pearson, Inc. Example: Graphing Sequences Slide 9.4 - 22

23. Copyright © 2011 Pearson, Inc. Example: Graphing Sequences Slide 9.4 - 23

24. Copyright © 2011 Pearson, Inc. Slide 9.4 - 24 The Fibonacci Sequence

25. Copyright © 2011 Pearson, Inc. Fibonacci in Nature Slide 9.4 - 25