1. 8.1 Sequences and Series
Essential Questions: How do we use sequence notation to write the terms of a sequence? How do we use factorial notation? How do we use summation notation to write sums?

2. Definition of a 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.

3. 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

4. Finding Terms of a Sequence
Write out the first five terms of the sequence given by
Solution:

5. Finding the nth term of a Sequence
Write an expression for the apparent nth term (an) of each sequence.
a. 1, 3, 5, 7, … b. 2, 5, 10, 17, …
Solution:
n: n
terms: an
n: … n
terms: … an
Apparent pattern: Each term is 1 less than twice n, which implies that
Apparent pattern: Each term is 1 more than the square of n, which implies that

6. 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

7. 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.

8. Finding the Terms of a Sequence Involving Factorials
List the first five terms of the sequence given by
Begin with n = 0.

9. Evaluating Factorial Expressions
Evaluate each factorial expression. Make sure you use parentheses when necessary.
a b c.
Solution: a. b. c.

10. 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?

11. 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:

12. 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.

13. Summation Notation for Sums
Find each sum.
a b c.
Solution: a.

14. 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.

15. How to Input Sums in your calculator
Be sure you are in sequence mode on the calculator!
Good news! This can all be done using the TI-84 Plus graphing calculator.
To enter in example a, hit the following keys:
The following screen will appear:
Now hit 5. Then, hit
Good news! This can all be done using the TI-84 Plus graphing calculator.
To enter in example a, hit the following keys:
The following screen will appear:
Now hit 5. Then, hit
The following screen should appear:
Choose 5.

16. This is what your calculator screen should look like:
Now type in the sum, the variable, the lower limit, the upper limit, and the increment (default is 1).

17. Assignment
Page
2-32 even, odd, even, even, even