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InfniteSeries
InfniteSeries
Module 1
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Learning Outcome
Learning Outcome
At the end of the module, students able to
identify the difference between sequenceand series.
determine whether a series converges or
diverges.
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An infinite sequence :
,...,...,,, 321 naaaa
What is an infnite sequence?
What is an infnite sequence?
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An infinite series is formed by addingterms
of an infinite sequence;
......321 +++++
n
aaaa
In other words:
In other words:
......3211
+++++=
=
n
n
n aaaaa
What is an infnite series?
What is an infnite series?
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Infnite sequence versus series
Infnite sequence versus series
sequence series
na
nS
1a 11 aS =
2a 212 aaS +=
3a3213 aaaS ++=
...3211
+++=
=
aaaa
n
n
Infinite Seriesor Series
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Example 1
Example 1
=
+++=1
...8
1
4
1
2
1
2
1
,...
2
1,...,
8
1,
4
1,
2
1
)i(
n
n
n sequence
series
1S
2S
3S
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Example 1Example 1
=
+
+
++++=+
+
1
1
1
...)4
1()
3
1(
2
11
1
)1(
,...1
)1(,...,4
1,3
1,2
1,1)ii(
k
k
k
k
k sequence
series
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Let denote the series nthpartial sum:
......3211
+++++=
=
n
n
n aaaaa
The series converge or diverge?
Given a series
nS
n
n
k
kn aaaaaS ++++==
=
...3211
321
3
1
3
21
2
1
2
11
aaaaS
aaaS
aS
n
n
n
n
++==
+==
=
=
=
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The series converges.
f the sequence !Sn" is convergent and
e#ists , then the series
is said to be convergent to S.
SSnn =lim
=1n
na
A sequence of partial sum
is formed.
,...,...,,, 321 nSSSS
$ $ $ $
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The series converges.
%he numberSis called the sum of the series,
and we write .1
San
n =
=
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The series diverges.
f does not e#ists,
the series is said to be divergent.
nn
S
lim
=1n
na
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o! to determine a seriesconverge"diverge?
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=
+++=1
...8
1
4
1
2
1
2
1
n
n
IllustrationIllustration
nnS
S
S
S
S
2
1
4
3
2
1
1
.
.
.
16
15
16
1
8
1
4
1
2
1
87
81
41
21
4
3
4
1
2
1
21
=
=+++=
=++=
=+=
=
&ow this formula
is obtained$
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IllustrationIllustration
nS,...,
8
7,
4
3,
2
1
Thus,
nnS
S
S
S
S
2
1
4
3
2
1
1
.
.
.
16
15
16
1
8
1
4
1
2
1
87
81
41
21
4
3
4
1
2
1
2
1
=
=+++=
=++=
=+=
=
n211,...,
211,
211,
211 32
nnS
2
11=
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==+++=
11...8
1
4
1
2
1
2
1
n
n
IllustrationIllustration
1)1(limlim2
1==
nnnn
S
%he series is convergent and has the sum
equal to ', i.e.,1
2
1
1
=
=nn
.
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IllustrationIllustration
.
n Sum o# the frst n terms $ %
1 &.'
( &.)'* &.+)', &.-*)'' &.-+)' &.-+,*)'
) &.--(1+)'+ &.--(1+)'&
=12
1
n
n
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(etermine whether the series
converges or diverges.
Example 2
=
1
)1(n
n
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Example 2
=
1
)1(n
n
1111
011
1
3
2
1
=+=
==
=
S
S
S
diverges.
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Exercises
Exercise 10.3 , !"e 653
#o. 1$2, 15$16
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E/0 o# O02LE
