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