Chapter 9.1 Sequences and Series

1. Chapter 9.1 Sequences and Series

2. Definition of Sequences • A sequence can be described as a function whose domain is the set of positive integers f(1) = a1, f(2) = a2, f(3) = a3, f(4) = a4, f(n) = an • Infinite Sequence • A function whose domain is the set of positive integers. The function values a1, a2, a3, a4, … an are the terms of the sequence • If the domain of the function consists of the first npositive integers only, the sequence is a finite sequence.

3. Writing the Terms of a Sequence • Write the first four terms of the sequences given by: a. an= 3n-2 b. an= 3 + (-1)n You Try: an = 3n + 1

4. A Sequence Whose Terms Alternate Sign • Write the first five terms of the sequence given by • You Try:

5. Finding the nth Term of a Sequence • Write an expression for the apparent nth term (an) of each sequence • 1, 3, 5, 7, . . . • 2, -5, 10, -17, . . . • You Try: 1, 4, 7, 10, 13, . . .

6. The Fibonacci Sequence: A Recursive Sequence • To define a sequence recursively, you need to be given one or more of the first few terms. All other terms of the sequence are then defined using previous terms. • Fibonacci was an Italian mathematician who wanted to study how rabbits populate. His hypothesis was: • “How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?”

7. Fibonacci Sequence cont. • Fibonacci found that rabbits can mature after one month and then begin repopulating. The numbers he calculated were as follows. • 1, 1, 2, 3, 5, 8, . . . • Can you figure out how he determined this sequence? Try to find the rule and write it on your paper.

8. Fibonacci Sequence Rule • The Fibonacci Sequence is defined recursively as follows. • a0 = 1 • a1 = 1 • ak = ak-2 + ak-1, where k ≥ 2

10. Factorials • If n is a positive integer, n factorial is defined as n! = 1  2  3  4  . . . (n-1) n. • As a special case, zero factorial is defined as 0! = 1 • Examples of the first several nonnegative integers: • 0! = 11! = 12! = 1  2 = 23! = 1 2 3 = 64! = 1 234 = 24 • Factorials follow the same order of operations as exponents. For example: 2n! = 2(n!) = 2(1234. . . n)

11. Writing the Terms of a Sequence Involving Factorials • Write the first five terms of the sequence given by • You try

12. Evaluating Factorial Expressions • Evaluate each factorial expression • a. • b.

13. Summation Notation • Summation notation involves the use of the uppercase Greek letter Σ (sigma). • The sum of the first n terms of a sequence is represented by • where iis called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation

14. Summation Notation for Sums • Find each sum. • You Try:

15. Properties of Sums p. 647 1. 2. 3. 4.

16. Series • The sum of the terms of a finite or infinite sequence is called a series. • Definition (finite sequence): the sum of the first n terms of the sequence is called a finite series or the nth partial sum of the sequence and is denoted by • a1 + a2 + a3 + . . . + an = • Definition (infinite sequence): the sum of all the terms of the infinite sequence is called an infinite series and is denoted by • a1 + a2 + a3 + . . . + ai =

17. Finding the Sum of a Series • For the series find (a) the third partial sum and (b) the sum. • You Try:Find the fourth partial sum