9.1 Sequences and 9.1 Sequences and SeriesSeries

Definition of SequenceDefinition of Sequence

An ordered list of numbersAn ordered list of numbers An An infinite sequenceinfinite sequence is a function whose is a function whose

domain is the set of positive integersdomain is the set of positive integers The function values aThe function values a11, a, a22, a, a33,…,a,…,ann,… are ,… are

the the termsterms of the sequence of the sequence A A finite sequencefinite sequence has a domain of the first has a domain of the first

n positive integers only; an positive integers only; a11, a, a22, …, a, …, ann

Recursively defined sequencesRecursively defined sequences

Sequences where one or more of the first Sequences where one or more of the first terms is giventerms is given

Write the first five terms; Write the first five terms; aa11=25 a=25 ak+1k+1= a= ak k – 5– 5

Factorial NotationFactorial Notation

If n is a positive integer then n!If n is a positive integer then n!=1=1∙2∙3∙4∙∙∙(n-1)∙n∙2∙3∙4∙∙∙(n-1)∙n

0!=1 by definition0!=1 by definition 6!= 6!= 11∙2∙3∙4∙5∙6=720∙2∙3∙4∙5∙6=720 2n!≠(2n)!2n!≠(2n)! On the calculator MATH PRB 4:! On the calculator MATH PRB 4:!

Summation NotationSummation Notation The sum of the first n terms of a sequence The sum of the first n terms of a sequence

is represented by is represented by

where where ii is the is the index of summationindex of summation, , nn is the is the upper limit upper limit and 1 is the and 1 is the lower limitlower limit

(lower limit does not have to be 1 and any letter can be used (lower limit does not have to be 1 and any letter can be used for the index of summation)for the index of summation)

n

n

ii aaaaa

...3211

Using the calculatorUsing the calculator

To sum the first n terms of a sequence; To sum the first n terms of a sequence; use the LIST menuuse the LIST menu 22ndnd STAT MATH 5:sum( 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))

functionvariable Lower limit

Upper limit

Properties of SumsProperties of Sums

n

i

n

i

n

i

n

ii

baba

acca

11

11

111

11

n

11

)( .2

.1

Definition of a SeriesDefinition of a Series

The sum of all terms of the infinite The sum of all terms of the infinite sequence is called the sequence is called the infinite seriesinfinite series

The sum of the first n terms of the The sum of the first n terms of the sequence is called a sequence is called a finite seriesfinite series or the or the nth partial sumnth partial sum of the sequenceof the sequence

1

321 ......i

ii aaaaa

n

iin aaaaa

1321 ...

Example: Write the first 5 termsExample: Write the first 5 terms

aa1 1 = 15 a = 15 ak+1k+1 = a = akk +3 +3

Example: Find the sumExample: Find the sum5

2

( 1)( 3)k

k k

Assignment:Assignment:

PAGE 625 #3-90 multiples of 3PAGE 625 #3-90 multiples of 3