1. Section 8.1 Sequences & Series

2. Sequences & Series Definition of Sequence: An infinite sequence is a function whose domain is the set of positive integers. The function values a 1, a 2, a 3, a 4, …, a n,… are the terms of the sequence. If the domain of a function consists of the first n positive integers, the sequence is a finite sequence.

3. Sequences & Series What are the first six terms of the sequence defined by a n = 4n – 3 a 1 = 4(1) – 3 = 1 a 2 = 4(2) – 3 = 5 a 3 = 4(3) – 3 = 9 a 4 = 4(4) – 3 = 13 a 5 = 4(5) – 3 = 17 a 6 = 4(6) – 3 = 21 a 1 means the first term. a 2 means the second term. a 3 means the third term. a 4 means the fourth term. a 5 means the fifth term. a 6 means the sixth term. The first six terms are 1, 5, 9, 13, 17, and 21.

4. Sequences & Series Write the first five terms of the sequence defined by a n = – 2 n 3n+1

5. Sequences & Series One of the best known sequences is the Fibonacci Sequence. Read about the Fibonacci Sequence at http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#rabeecow Click on Rabbits, Cows, and Bees Family Trees What are the first 12 term of the Fibonacci Sequence? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144.

6. Sequences & Series Factorial Notation Definition of Factorial If n is a positive integer, n factorial is defined as n! = n · (n – 1) · (n – 2) · · · 4 · 3 · 2 · 1 As a special case, zero factorial is defined as 0! = 1. Ex: What is the value of 6! 6! = 1 · 2 · 3 · 4 · 5 · 6 = 720 Can you find the factorial symbol on the calculator?

7. Sequences & Series Evaluate the following:

8. Sequences & Series 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 called the lower limit of summation.

9. Sequences & Series Examples:

10. Sequences & Series Series Definition of a Series Consider the infinite sequence a 1, a 2, a 3, …, a n, … 1.The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by 2.The sum of all the terms of the infinite sequence is called an infinite series and is denoted by

11. Sequences & Series Examples: 1. Find the fourth partial sum of the series:

12. Sequences & Series 2. Find the sum of the series : Example: Note: As more and more terms are added, the closer the sum approaches 3 although it will never exactly equal 3. This will be discussed later.

13. Sequences & Series Example: Rosemary deposited $8000 into an account that earns 4.5% interest compounded quarterly. The balance in her account after n quarters is given by: 1. Determine the first five terms of the sequence. 2. Find the balance in this account after 5 years

14. Sequences & Series Answers to previous example: 1.2.

15. Sequences & Series What you should know: 1. Sequence notation and how to determine the terms of the sequence. 2. The meaning of factorial notation and how to simplify. 3. Summation notation and how to expand to find a sum. 4. Find the sum of a series