1. Section 9.1
Sequences and Series

2. Objective By following instructions students will be able to:
Use sequences notation to write the terms of sequences.
Use factorial notation.
Use summative notation to write sums.
Find sums of infinite series.
Use sequences and series to model and solve real-life problems..

3. Definition of Sequence
An infinite sequence is a function whose domain is the set of positive integers. The function values
are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

4. Example 1:
Find the first four terms of the sequences by a) b)

5. Example 2:
Find the first five terms of the sequence given by

6. U-TRY #1
Write the first five terms of the sequence. (Assume n begins with 1). a) b)

7. Example 3:
Write an expression for the apparent nth term of each sequence.
1, 3, 5, 7, …….
2, 5, 10, 17, …….

8. Example 4:
The Fibonacci sequence is defined recursively as follows. Where . Write the first six terms of this sequence.

9. Definition of Factorial
If n is a positive integer, n factorial is defined by
As a special case, zero factorial is defined by 0!=1.
Note: Factorials follow the same conventions for order of operations as do exponents.
Ex.

10. Example 5:
List the first five terms of the sequence given by . Begin with .

11. Example 6:
Evaluate each factorial expression. a) b) c)

12. U-TRY #2
Find the indicated term of the sequence. a) b)

13. Definition of Summative 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.

14. Example 7:
Evaluate. a) b) c)

15. Properties of Sums 1)
where c is any constant. 2) 3)

16. Definition of a Series
Consider the infinite series
The sum of all terms of the infinite sequence is called an infinite series and is denoted by

17. Definition of a Series
Consider the infinite series
2. 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

18. Example 8:
For the series , find
the 3rd partial sum.
the sum.

19. Example 9:
From 1960 to 1997, the resident population of the United States can be approximated by the model Where is the population in millions and n represents the calendar year, with n=0 corresponding to Find the last five terms of this finite sequence.

20. U-TRY #3
Find the sum. a) b)

21. Revisit Objective Did we…
Use sequences notation to write the terms of sequences?
Use factorial notation?
Use summative notation to write sums?
Find sums of infinite series?
Use sequences and series to model and solve real-life problems?

22. Homework
Pg 625 #s 1-33 ODD, 51, 57, 61, EOO, 109