The purpose of this section is to discuss sums that contains infinitely many terms

The purpose of this section is to discuss sums that contains infinitely many terms

For example For example When we write 6

1

in the decimal form 6666.06

1 We mean

32 )10(

6

)10(

6

10

6

6

1

which suggests that the decimal representation of 6

1

can be viewed as a sum of many real numbers .

The most familiar example of such sums occur in the decimal representation of real numbers.

The most familiar example of such sums occur in the decimal representation of real numbers.

The numbers ,3

,2

,1

uuu are called the term of the series

An infinite series is an expression that can be written in the form

kuuu

kku

211

Definition (1)Definition (1)

and

n

kkusn1

is called the nth partial sum of series

1kku

and the sequence

is called the sequence of partial sums

1nns

Definition (2) Definition (2)

be the sequence of partial sums of seriesLet

kuuu

kku

211

The series

1kku

is said to converge to a number s iff

n

kkus

nn

ns

1limlim In which case we call s

the sum of the series and write .1

sk

ku

If no such limit exists the series is said to diverge

1nns

Example 1Example 1 Determine whether the series

222222converge or diverge .If it is converge, find the sum

Solution: Solution:

,21s 0222 s22223 s and so on

the sequence of partial sums is ,0,2,0,2Since this is divergent sequence and so the given series diverges

and consequently no sum

Definition (3) Definition (3) A series of the form

132 nararararais called a geometric series

Here are some examples

)3,1(3931 rak

)2

1,

2

1(

2

1)1(

8

1

4

1

2

1 1

rak

k

)1,1(1111 ra

Theorem 1Theorem 1 A geometric series

0,32

0

aararararaar k

k

k

Converges if 1r and diverge if 1r

the sum is r

aar

k

k

10

Proof: Proof:

First , if 1r then the series is

aa ansn )1(

ansn

nn

)1(limlim

If the series converges ,then

Also, if then the series is

aa1r

the sequence of partial sums is

,0,,0, aa Which diverges

Moreover , 1rn

n ararararas 32

12 nnn ararararrs

1 nnn ararss

r

ar

r

a

r

aras

nn

n

111

11

r

asn

n

1lim

Similarly for 1r

Example 2Example 2 Determine whether the following series

converge or diverge .If it is converge, find the sum

a-

0 2

3

kk

b-k

k

k

1

1

2 75 c-

1k

kx

Solution: Solution:

0 2

3

kk

is a converge geometric series with 2

1,3 ra

and the sum is 6

21

1

3

1

ra

b-

1

1

11

11

21

1

2

)7

25(25

7

25

7

575

k

k

kk

k

kk

kk

k

k

the series diverges

c-

0k

kx the series is geometric series with xra ,1

If the series is converges and1r then

and diverges otherwise

xx

k

k

1

1

0

Exercise: Find the rational number represented by the repeating decimal

153153153.0

Evaluate Evaluate k

kk

1 3

2

Solution: Solution:

kk

k 1 3

2

32

3

23

3

22

3

2

32

3

2

3

2

3

2

32

3

2

3

2

3

3

2

6

32

1

2

9

8

3

42

32

1

32

32

1

32

32

1

32 32

kk

k 1 3

2

Creative thinkingCreative thinking

Worksheet Worksheet

Find the value to which each the following series converges1 -

a- d-c-b-

e- f-

0 )1(

1

k kk g-k

k

)3(.21

2 -A ball is dropped from height of 10m.Each time it strikes the

ground it bounces vertically to height that is

of the preceding height. Find the total distance the ball will travel

if it is assumed to bounce infinitely often.

5

4

Answer=90m