9.1 Sequences and Series

Definition of Sequence  An ordered list of numbers  An infinite sequence is a function whose domain is the set of positive integers  The function values a 1, a 2, a 3,…,a n,… are the terms of the sequence  A finite sequence has a domain of the first n positive integers only; a 1, a 2, …, a n

Recursively defined sequences  Sequences where one or more of the first terms is given  Write the first five terms; a 1 =25 a k+1 = a k – 5 a 1 =25 a k+1 = a k – 5

Factorial Notation  If n is a positive integer then n!=1∙2∙3∙4∙∙∙(n-1)∙n  0!=1 by definition  6!= 1∙2∙3∙4∙5∙6=720  2n!≠(2n)!  On the calculator MATH PRB 4:!

Summation Notation  The sum of the first n terms of a sequence is represented by where i is the index of summation, n is the upper limit and 1 is the lower limit (lower limit does not have to be 1 and any letter can be used for the index of summation)

Using the calculator  To sum the first n terms of a sequence; use the LIST menu 2 nd STAT MATH 5:sum( 2 nd STAT MATH 5:sum( 2nd STAT OPS 5:seq( 2nd STAT OPS 5:seq( sum(seq(1/n!, n, 0, 8)) sum(seq(1/n!, n, 0, 8)) function variable Lower limit Upper limit

Properties of Sums

Definition of a Series  The sum of all terms of the infinite sequence is called the infinite series  The sum of the first n terms of the sequence is called a finite series or the nth partial sum of the sequence

Example: Write the first 5 terms  a 1 = 15 a k+1 = a k +3

Example: Find the sum

Assignment:  PAGE 625 #3-90 multiples of 3