11-4 INTRO TO SERIES

• DEFINITION

• A SERIES IS THE SUM OF THE TERMS OF A SEQUENCE.

• SEQUENCE VS. SERIES

2, 4, 8, … 2 + 4 + 8 + …

NTH PARTIAL SUM

•THE NTH PARTIAL SUM IS THE SUM OF THE FIRST N TERMS OF THAT SERIES.

• SN= SUM OF THE FIRST N

TERMS

SIGMA NOTATION

New notation: ∑ sigma = the sum of

upper limit

lower limit

tn FORMULA

6

1

(2 1)k

k

= 1 + 3 + 5 + 7 + 9 + 11 = 36

TOO

• EVALUATE THE EXPRESSION:4

1

3k

k

3(1) + 3(2) + 3(3) + 3(4) =

3 + 6 + 9 + 12 = 30

EX 2) EVALUATE THE EXPRESSION

= (-1)1(1+2)

3

1

( 1) ( 2)k

k

k

+ (-1)2(2+2)+ (-1)3(3+2)

= (-1)(3) + (1)(4)+ (-1)(5)

= – 3 + 4 – 5 = -4

SIGMA NOTATION• WE WANT:

• WHERE TK IS THE FORMULA TO FIND THE VALUE OF THE

NTH TERM.

ARITHMETIC: TK = T1 + (N-1)D

GEOMETRIC: TK = T1(R)N-1

1

n

kk

t

HOW TO WRITE SIGMA NOTATION

• WRITE SN USING SIGMA NOTATION

• EXAMPLE:

S30 FOR 1 + 8 + 27 + 64 + 125 + …

IS THE SERIES ARITHMETIC, GEOMETRIC, OR NEITHER?

NEITHER!!!!!

SIGMA NOTATION

• LOOK FOR A PATTERN FOR TK

S30 FOR 1 + 8 + 27 + 64 + 125 + …

• TK = K3

SO, OUR ANSWER IS:

303

1k

k

tn = 10 + (n – 1)(5)

EX USE SIGMA NOTATION TO WRITE THE SERIES

10 + 15 + 20 + … + 100

tn = 10 + 5n – 5 = 5n + 5

19

1

5 5k

k

Find upper limit: (what term # is 100)tn = 5n + 5 100 = 5n + 5 19 = n

Arithmetic d = 5 t1 = 10

Series is geometric!!

tn= t1 (r)n-1

EX USE SIGMA NOTATION TO WRITE THE SERIES

t1= 1 r = - 1/3

4

113

1

1k

k

1 1 113 9 27

How did we get r?

Check it!!!

113

1 3

HOMEWORK

PG. 579-581 Q1-10 #1 – 35 ODD