1. 12.1 Sequences and Series ©2001 by R. Villar All Rights Reserved

2. Sequences and Series Sequence: function whose domain is the set of positive integers. a 1, a 2, a 3, a 4,..., a n,... are the terms of the sequence. If the domain only contains n positive integers, the sequence is a finite sequence.

3. Example: Find the first five terms of the sequence a n = 2n – 3 a 1 = 2(1) – 3 = –1 a 2 = 2(2) – 3 = 1 a 3 = 2(3) – 3 = 3 a 4 = 2(4) – 3 = 5 a 5 = 2(5) – 3 = 7 {–1, 1, 3, 5, 7} A recursive function is a function whose domain is the set of non-negative integers. A well known recursive function is the factorial function. (We use an exclamation mark to denote a factorial)...

4. The factorial is defined as follows: n! = n (n – 1) 0! = 1 1! = 1 1 = 1 2! = 1 1 2 = 2 3! = 1 1 2 3 = 6 4! = 1 1 2 3 4 = 24 5! = 1 1 2 3 4 5 = 120 Example: Find the first four terms of the sequence a n = n! + 1 a 1 = 1 + 1 = 2 a 2 = 2 1 + 1 = 3 a 3 = 3 2 1 + 1 = 7 a 4 = 4 3 2 1 + 1= 25 {2, 3, 7, 25}

5. To write the sum of a finite sequence, we can use summation notation. This is also called Sigma Notation. Here’s an example of an expression in Sigma notation: Greek letter Sigma The expression formed by adding the first n terms of a sequence is called a series.

6. Example: Write the series represented by the summation notation. Then find the sum. The notation means “The sum from n = 1 to 5 of 2n” = 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2 + 4 + 6 + 8 + 10 = 30

7. Ex. Write the series represented by the summation notation. Then find the sum. = 12 + 12 + 12 + 12 0! 1! 2! 3! = 12 + 12 + 12 + 12 1 1 2 6 = 32