1. Copyright © 2007 Pearson Education, Inc. Slide 8-1 Warm-Up Find the next term in the sequence: 1, 1, 2, 6, 24, 120,…

2. Copyright © 2007 Pearson Education, Inc. Slide 8-2 Warm-Up Find the first four terms of the sequence given by

3. Copyright © 2007 Pearson Education, Inc. Slide 8-3 Chapter 8: Sequences and Series 2015

4. Copyright © 2007 Pearson Education, Inc. Slide 8-4 Chapter 8: Sequences, Series, and Probability 8.1Sequences and Series 8.2Arithmetic Sequences and Series 8.3Geometric Sequences and Series 8.4 Mathematical Induction 8.5 The Binomial Theorem

5. Copyright © 2007 Pearson Education, Inc. Slide 8-5 8.1 Sequences Sequences are ordered lists generated by a function, for example f(n) = 100n

6. Copyright © 2007 Pearson Education, Inc. Slide 8-6 8.1 Sequences f (x) notation is not used for sequences. Write Sequences are written as ordered lists a 1 is the first element, a 2 the second element, and so on A sequence is a function that has a set of natural numbers as its domain.

7. Copyright © 2007 Pearson Education, Inc. Slide 8-7 8.1 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 Solution

8. Copyright © 2007 Pearson Education, Inc. Slide 8-8 8.1 Graphing Sequences The graph of a sequence, a n, is the graph of the discrete points (n, a n ) for n = 1, 2, 3, … Example Graph the sequence a n = 2n. Solution

9. Copyright © 2007 Pearson Education, Inc. Slide 8-9 8.1 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, …

10. Copyright © 2007 Pearson Education, Inc. Slide 8-10 8.1 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.

11. Copyright © 2007 Pearson Education, Inc. Slide 8-11 8.1 Convergent and Divergent Sequences Example : Find the first 5 terms of the sequence. Is the sequence convergent or divergent?

12. Copyright © 2007 Pearson Education, Inc. Slide 8-12 8.1 Convergent and Divergent Sequences Solution: The sequence converges to 0. The terms of the sequence 1, 0.5, 0.33.., 0.25, … grow smaller and smaller approaching 0. This can be seen graphically.

13. Copyright © 2007 Pearson Education, Inc. Slide 8-13 8.1 Convergent and Divergent Sequences Example : Find the first 6 terms of the sequence. Is the sequence convergent or divergent?

14. Copyright © 2007 Pearson Education, Inc. Slide 8-14 8.1 Convergent and Divergent Sequences Solution: The sequence is divergent. The terms grow large without bound 1, 4, 9, 16, 25, 36, 49, 64, … and do not approach any one number.

15. Copyright © 2007 Pearson Education, Inc. Slide 8-15 8.1 Convergent and Divergent Sequences Example Is the sequence convergent or divergent? Solution: The sequence converges to 2/3

16. Copyright © 2007 Pearson Education, Inc. Slide 8-16 Finding Terms of a Sequence The first four terms of the sequence given by are: The first four terms of the sequence given by are:

17. Copyright © 2007 Pearson Education, Inc. Slide 8-17 Finding Terms of a Sequence Write out the first five terms of the sequence given by Solution:

18. Copyright © 2007 Pearson Education, Inc. Slide 8-18 Finding the nth term of a Sequence Write an expression for the apparent nth term (a n ) of each sequence. a. 1, 3, 5, 7, … b. 2, 5, 10, 17, … Solution: a. n: 1 2 3 4... n terms: 1 3 5 7... a n Apparent pattern: Each term is 1 less than twice n, which implies that b. n: 1 2 3 4 … n terms: 2 5 10 17 … a n Apparent pattern: Each term is 1 more than the square of n, which implies that

19. Copyright © 2007 Pearson Education, Inc. Slide 8-19 Additional Example Write an expression for the apparent nth term of the sequence: Solution: Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that

20. Copyright © 2007 Pearson Education, Inc. Slide 8-20 Factorial Notation If n is a positive integer, n factorial is defined by As a special case, zero factorial is defined as 0! = 1. Here are some values of n! for the first several nonnegative integers. Notice that 0! is 1 by definition. The value of n does not have to be very large before the value of n! becomes huge. For instance, 10! = 3,628,800.

21. Copyright © 2007 Pearson Education, Inc. Slide 8-21 Finding the Terms of a Sequence Involving Factorials List the first five terms of the sequence given by Begin with n = 0.

22. Copyright © 2007 Pearson Education, Inc. Slide 8-22 Evaluating Factorial Expressions Evaluate each factorial expression. Make sure you use parentheses when necessary. a. b. c. a. b. c.

23. Copyright © 2007 Pearson Education, Inc. Slide 8-23 Additional Example Write an expression for the apparent nth term of the sequence: Solution: Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that

24. Copyright © 2007 Pearson Education, Inc. Slide 8-24 Have you ever seen this sequence before? 1, 1, 2, 3, 5, 8 … Can you find the next three terms in the sequence? Hint: 13, 21, 34 Can you explain this pattern?

25. Copyright © 2007 Pearson Education, Inc. Slide 8-25 The Fibonacci Sequence Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. A well-known example is the Fibonacci Sequence. The Fibonacci Sequence is defined as follows: Write the first six terms of the Fibonacci Sequence:

26. Copyright © 2007 Pearson Education, Inc. Slide 8-26 Example Write the first five terms of the recursively defined sequence: Solution: 5, 8, 11, 14, 17

27. Copyright © 2007 Pearson Education, Inc. Slide 8-27 Homework Day 1: Pg. 563 1-9odd, 21-23odd, 35-69 odd Day 2: 71-81 odd, 91-103 odd

28. Copyright © 2007 Pearson Education, Inc. Slide 8-28 HWQ Write an expression for the apparent nth term of the sequence.

29. Copyright © 2007 Pearson Education, Inc. Slide 8-29 8.1 Day 2 Series 2015

30. Copyright © 2007 Pearson Education, Inc. Slide 8-30 Summation Notation Definition of Summation Notation The sum of the first n terms of a sequence is represented by Where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation.

31. Copyright © 2007 Pearson Education, Inc. Slide 8-31 8.1 Series and Summation Notation A finite series is an expression of the form and an infinite series is an expression of the form.

32. Copyright © 2007 Pearson Education, Inc. Slide 8-32 Summation Notation for Sums Find each sum. a. b. c. Solution: a.

33. Copyright © 2007 Pearson Education, Inc. Slide 8-33 Solutions continued b. c. Notice that this summation is very close to the irrational number. It can be shown that as more terms of the sequence whose nth term is 1/n! are added, the sum becomes closer and closer to e.

34. Copyright © 2007 Pearson Education, Inc. Slide 8-34 8.1 Series and Summation Notation Summation Properties If a 1, a 2, a 3, …, a n and b 1, b 2, b 3, …, b n are two sequences, and c is a constant, then for every positive integer n, (a)(b) (c)

35. Copyright © 2007 Pearson Education, Inc. Slide 8-35 8.1 Series and Summation Notation Summation Rules These summation rules can be proven by mathematical induction.

36. Copyright © 2007 Pearson Education, Inc. Slide 8-36 8.1 Series and Summation Notation Example Use the summation properties to evaluate (a) (b) (c) Solution (a)

37. Copyright © 2007 Pearson Education, Inc. Slide 8-37 8.1 Series and Summation Notation (b) (c) (b) (c)

38. Copyright © 2007 Pearson Education, Inc. Slide 8-38 Homework Day 1: Pg. 563 1-9odd, 21-23odd, 35-69 odd Day 2: 71-81 odd, 91-103 odd