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Page 1
For more notes, call 8130648819
INFINITE SERIES
Topics
1. Definition of infinite series, sequence of partial sums.
2. Cauchy criterion
3. Boundedness of sequence of partial sums,
4. Comparison test, limit comparison tests
5. D’ Alembert’s Ratio test, Cauchy’s nth root test
6. Raabe’s test, Logarithm test, Gauss test
7. Integral test (without proof)
8. Cauchy’s Condensation text
9. Alternating series, Leibniz test.
10. Absolute and Conditional convergence
In elementary texts,an infinite series is sometimes defined to be an expression of the form x
x
x
However,this definition lac s clarity,since there is a priori no particular value that we can attach to this array of
symbols,which call for an infinite number of additions to be performed
Def
If X (x
) is a sequence in ,then the infinite series (or simply the series) generated by X is the sequence
S (s
) defined by
s
x
s
s
x
( x
x
)
s
s
x
( x
x
x
)
The numbers x
are called the terms of the series and the numbers s
are called the partial sums of this series If limS
exits,(limit of sequence of partial sum) we say that this series is convergent and call this limit the sum or the value of
this series If this limit does not exists,we say that the series S is divergent
It is convenient to use symbols such as
( ) ?(x
) or ?x
or ? x
As a sequence may be indexed such that its first element is not x
,but is x
,or x
or x
,we will denote the series having
these numbers as their first element by the symbols
? x
or ? x
or ? x
Partial Sums
S
u
S
u
u
S
u
u
u
u
are called the first,second, ,nth partial sums of the series of ?u
Note To every infinite series u
there corresponds a sequence S
of its partial sums It should be noted that
when the first term in the series is x
,then the first partial sum is denoted by s
Warning The reader should guard against confusing the words sequence’’ and series’’ In nonmathematical language,
these words are interchangeable however,in mathematics these words are not synonyms Indeed,a series is a
sequence S (s
) obtained from a given sequence X (x
) according to the special procedure given in
Definition
series of positive terms
Def
If all the terms of the series u
u
u
u
are positive i e ,if u
n,then the
series u
is called a series of positive terms
Alternating series
Def
A series in which the terms are alternatively positive and negative is called an alternating series
Thus,the series ?( )
u
u
u
u
u
( )
u
where u
n is an alternating series
ehaviour of an Infinite Series
Def
An infinite series, u
converges,diverges or oscillates (finitely or infinitely) according as the sequence S
of its partial sums converges,diverges or oscillates (finitely or infinitely)
(i) The series u
converges (or is said to be convergent) if the sequence S
of its partial sums converges
Thus, u
is convergent if lim
S
Finite
(ii) The series u
diverges (or is said to be divergent) if the sequence S
of its partial sums diverges
Thus, u
is divergent if lim
S
or
(iii) The series u
oscillates finitely if the sequence S
of its partial sums oscillates finitely
Thus, u
oscillates finitely if S
is bounded and neither converges nor diverges
Page 2
For more notes, call 8130648819
INFINITE SERIES
Topics
1. Definition of infinite series, sequence of partial sums.
2. Cauchy criterion
3. Boundedness of sequence of partial sums,
4. Comparison test, limit comparison tests
5. D’ Alembert’s Ratio test, Cauchy’s nth root test
6. Raabe’s test, Logarithm test, Gauss test
7. Integral test (without proof)
8. Cauchy’s Condensation text
9. Alternating series, Leibniz test.
10. Absolute and Conditional convergence
In elementary texts,an infinite series is sometimes defined to be an expression of the form x
x
x
However,this definition lac s clarity,since there is a priori no particular value that we can attach to this array of
symbols,which call for an infinite number of additions to be performed
Def
If X (x
) is a sequence in ,then the infinite series (or simply the series) generated by X is the sequence
S (s
) defined by
s
x
s
s
x
( x
x
)
s
s
x
( x
x
x
)
The numbers x
are called the terms of the series and the numbers s
are called the partial sums of this series If limS
exits,(limit of sequence of partial sum) we say that this series is convergent and call this limit the sum or the value of
this series If this limit does not exists,we say that the series S is divergent
It is convenient to use symbols such as
( ) ?(x
) or ?x
or ? x
As a sequence may be indexed such that its first element is not x
,but is x
,or x
or x
,we will denote the series having
these numbers as their first element by the symbols
? x
or ? x
or ? x
Partial Sums
S
u
S
u
u
S
u
u
u
u
are called the first,second, ,nth partial sums of the series of ?u
Note To every infinite series u
there corresponds a sequence S
of its partial sums It should be noted that
when the first term in the series is x
,then the first partial sum is denoted by s
Warning The reader should guard against confusing the words sequence’’ and series’’ In nonmathematical language,
these words are interchangeable however,in mathematics these words are not synonyms Indeed,a series is a
sequence S (s
) obtained from a given sequence X (x
) according to the special procedure given in
Definition
series of positive terms
Def
If all the terms of the series u
u
u
u
are positive i e ,if u
n,then the
series u
is called a series of positive terms
Alternating series
Def
A series in which the terms are alternatively positive and negative is called an alternating series
Thus,the series ?( )
u
u
u
u
u
( )
u
where u
n is an alternating series
ehaviour of an Infinite Series
Def
An infinite series, u
converges,diverges or oscillates (finitely or infinitely) according as the sequence S
of its partial sums converges,diverges or oscillates (finitely or infinitely)
(i) The series u
converges (or is said to be convergent) if the sequence S
of its partial sums converges
Thus, u
is convergent if lim
S
Finite
(ii) The series u
diverges (or is said to be divergent) if the sequence S
of its partial sums diverges
Thus, u
is divergent if lim
S
or
(iii) The series u
oscillates finitely if the sequence S
of its partial sums oscillates finitely
Thus, u
oscillates finitely if S
is bounded and neither converges nor diverges
For more notes, call 8130648819
(iv) The series u
oscillates infitely if the sequence S
of its partial sums oscillates infinitely
Thus, u
oscillates infinitely if S
is unbounded and neither converges nor diverges
Ex Discuss the convergence of the series ?
n(n )
n(n )
to
Sol
Here u
n(n )
n
n
Putting n , , , ,n
u
, u
, u
, ,u
n
n
Now S
u
u
u
u
S
n
lim
S
S
converges to ?u
converges to
(Note For another method,see Comparisan Test)
Ex Show that the series
n
diverges to
Sol
S
n
n(n )( n )
lim
S
S
diverges to
The given series diverges to
Ex Show that the series n diverges to
Sol
S
n ( n)
n(n )
lim
S
S
diverges to
The given series diverges to
Ex Test for convergence of the series
(i) ?( )
n
(ii) ? sin.
n
/
Sol
(i) Here ?( )
n
S
, S
, S
S
, S
,S
etc
S
, , , , , , ,which is not bounded
S
is not convergent
?( )
n is not convergent
(ii) ? sin
n
v
v
v
v
v
v
S
v
, S
v
v
v ,
S
v
v
v , S
v
v
v
v
etc
S
v
,v ,v ,
v
, , ,
v
,v
Clearly,lim supS
v
and lim inf S
It follows that S
is not convergent
Hence,the given series is not convergent
Ex Prove that the series ?
converges to
Sol
We have
S
Page 3
For more notes, call 8130648819
INFINITE SERIES
Topics
1. Definition of infinite series, sequence of partial sums.
2. Cauchy criterion
3. Boundedness of sequence of partial sums,
4. Comparison test, limit comparison tests
5. D’ Alembert’s Ratio test, Cauchy’s nth root test
6. Raabe’s test, Logarithm test, Gauss test
7. Integral test (without proof)
8. Cauchy’s Condensation text
9. Alternating series, Leibniz test.
10. Absolute and Conditional convergence
In elementary texts,an infinite series is sometimes defined to be an expression of the form x
x
x
However,this definition lac s clarity,since there is a priori no particular value that we can attach to this array of
symbols,which call for an infinite number of additions to be performed
Def
If X (x
) is a sequence in ,then the infinite series (or simply the series) generated by X is the sequence
S (s
) defined by
s
x
s
s
x
( x
x
)
s
s
x
( x
x
x
)
The numbers x
are called the terms of the series and the numbers s
are called the partial sums of this series If limS
exits,(limit of sequence of partial sum) we say that this series is convergent and call this limit the sum or the value of
this series If this limit does not exists,we say that the series S is divergent
It is convenient to use symbols such as
( ) ?(x
) or ?x
or ? x
As a sequence may be indexed such that its first element is not x
,but is x
,or x
or x
,we will denote the series having
these numbers as their first element by the symbols
? x
or ? x
or ? x
Partial Sums
S
u
S
u
u
S
u
u
u
u
are called the first,second, ,nth partial sums of the series of ?u
Note To every infinite series u
there corresponds a sequence S
of its partial sums It should be noted that
when the first term in the series is x
,then the first partial sum is denoted by s
Warning The reader should guard against confusing the words sequence’’ and series’’ In nonmathematical language,
these words are interchangeable however,in mathematics these words are not synonyms Indeed,a series is a
sequence S (s
) obtained from a given sequence X (x
) according to the special procedure given in
Definition
series of positive terms
Def
If all the terms of the series u
u
u
u
are positive i e ,if u
n,then the
series u
is called a series of positive terms
Alternating series
Def
A series in which the terms are alternatively positive and negative is called an alternating series
Thus,the series ?( )
u
u
u
u
u
( )
u
where u
n is an alternating series
ehaviour of an Infinite Series
Def
An infinite series, u
converges,diverges or oscillates (finitely or infinitely) according as the sequence S
of its partial sums converges,diverges or oscillates (finitely or infinitely)
(i) The series u
converges (or is said to be convergent) if the sequence S
of its partial sums converges
Thus, u
is convergent if lim
S
Finite
(ii) The series u
diverges (or is said to be divergent) if the sequence S
of its partial sums diverges
Thus, u
is divergent if lim
S
or
(iii) The series u
oscillates finitely if the sequence S
of its partial sums oscillates finitely
Thus, u
oscillates finitely if S
is bounded and neither converges nor diverges
For more notes, call 8130648819
(iv) The series u
oscillates infitely if the sequence S
of its partial sums oscillates infinitely
Thus, u
oscillates infinitely if S
is unbounded and neither converges nor diverges
Ex Discuss the convergence of the series ?
n(n )
n(n )
to
Sol
Here u
n(n )
n
n
Putting n , , , ,n
u
, u
, u
, ,u
n
n
Now S
u
u
u
u
S
n
lim
S
S
converges to ?u
converges to
(Note For another method,see Comparisan Test)
Ex Show that the series
n
diverges to
Sol
S
n
n(n )( n )
lim
S
S
diverges to
The given series diverges to
Ex Show that the series n diverges to
Sol
S
n ( n)
n(n )
lim
S
S
diverges to
The given series diverges to
Ex Test for convergence of the series
(i) ?( )
n
(ii) ? sin.
n
/
Sol
(i) Here ?( )
n
S
, S
, S
S
, S
,S
etc
S
, , , , , , ,which is not bounded
S
is not convergent
?( )
n is not convergent
(ii) ? sin
n
v
v
v
v
v
v
S
v
, S
v
v
v ,
S
v
v
v , S
v
v
v
v
etc
S
v
,v ,v ,
v
, , ,
v
,v
Clearly,lim supS
v
and lim inf S
It follows that S
is not convergent
Hence,the given series is not convergent
Ex Prove that the series ?
converges to
Sol
We have
S
For more notes, call 8130648819
[ .
/
]
4 S
a( r
)
r
5
S
[
]
lim
[
]
The sequence S
converges to
?u
converges to
Hence,the given series converges to
Article The geometric series x x
x
to
(i) converges if x i e |x| and converges to
( )
(ii) diverges if x
Proof (i) When |x|
Since |x| x
as n
Now
S
x x
x
to n terms
( x
)
x
x
x
x
lim
S
x
lim
4
x
x
5
lim
S
x
0 lim
r
,if |r| 1
the sequence S
is convergent
the given series is convergent
(ii) When x
Sub case I When x
S
to n terms n
lim
S
the sequence S
diverges to
the given series diverges to
Sub case II When x , x
as n
S
x x
to n terms
(x
)
x
lim
S
lim
4
x
x
5
x
lim
S
the sequence S
diverges to
the given series diverges to
Illustrations
?
n
converge , p -
?
n
diverge , p -
?
vn
v
v
diverge [ p
]
?
n
is convergent [ p
]
Note A geometric series converges only when absolute value of its common ratio is numerically less than
(b) Consider the series generated by (( )
)
that is,the series
( ) ?( )
( ) ( ) ( )
It is easily seen (by mathematical induction) that s
if n is even and s
if n is odd therefore,the
sequence of partial sums is ( , , , , ) Since this sequence is not convergent,the series ( ) is divergent
Ex Examine the convergence of the series
(i)
to (ii)
to
Page 4
For more notes, call 8130648819
INFINITE SERIES
Topics
1. Definition of infinite series, sequence of partial sums.
2. Cauchy criterion
3. Boundedness of sequence of partial sums,
4. Comparison test, limit comparison tests
5. D’ Alembert’s Ratio test, Cauchy’s nth root test
6. Raabe’s test, Logarithm test, Gauss test
7. Integral test (without proof)
8. Cauchy’s Condensation text
9. Alternating series, Leibniz test.
10. Absolute and Conditional convergence
In elementary texts,an infinite series is sometimes defined to be an expression of the form x
x
x
However,this definition lac s clarity,since there is a priori no particular value that we can attach to this array of
symbols,which call for an infinite number of additions to be performed
Def
If X (x
) is a sequence in ,then the infinite series (or simply the series) generated by X is the sequence
S (s
) defined by
s
x
s
s
x
( x
x
)
s
s
x
( x
x
x
)
The numbers x
are called the terms of the series and the numbers s
are called the partial sums of this series If limS
exits,(limit of sequence of partial sum) we say that this series is convergent and call this limit the sum or the value of
this series If this limit does not exists,we say that the series S is divergent
It is convenient to use symbols such as
( ) ?(x
) or ?x
or ? x
As a sequence may be indexed such that its first element is not x
,but is x
,or x
or x
,we will denote the series having
these numbers as their first element by the symbols
? x
or ? x
or ? x
Partial Sums
S
u
S
u
u
S
u
u
u
u
are called the first,second, ,nth partial sums of the series of ?u
Note To every infinite series u
there corresponds a sequence S
of its partial sums It should be noted that
when the first term in the series is x
,then the first partial sum is denoted by s
Warning The reader should guard against confusing the words sequence’’ and series’’ In nonmathematical language,
these words are interchangeable however,in mathematics these words are not synonyms Indeed,a series is a
sequence S (s
) obtained from a given sequence X (x
) according to the special procedure given in
Definition
series of positive terms
Def
If all the terms of the series u
u
u
u
are positive i e ,if u
n,then the
series u
is called a series of positive terms
Alternating series
Def
A series in which the terms are alternatively positive and negative is called an alternating series
Thus,the series ?( )
u
u
u
u
u
( )
u
where u
n is an alternating series
ehaviour of an Infinite Series
Def
An infinite series, u
converges,diverges or oscillates (finitely or infinitely) according as the sequence S
of its partial sums converges,diverges or oscillates (finitely or infinitely)
(i) The series u
converges (or is said to be convergent) if the sequence S
of its partial sums converges
Thus, u
is convergent if lim
S
Finite
(ii) The series u
diverges (or is said to be divergent) if the sequence S
of its partial sums diverges
Thus, u
is divergent if lim
S
or
(iii) The series u
oscillates finitely if the sequence S
of its partial sums oscillates finitely
Thus, u
oscillates finitely if S
is bounded and neither converges nor diverges
For more notes, call 8130648819
(iv) The series u
oscillates infitely if the sequence S
of its partial sums oscillates infinitely
Thus, u
oscillates infinitely if S
is unbounded and neither converges nor diverges
Ex Discuss the convergence of the series ?
n(n )
n(n )
to
Sol
Here u
n(n )
n
n
Putting n , , , ,n
u
, u
, u
, ,u
n
n
Now S
u
u
u
u
S
n
lim
S
S
converges to ?u
converges to
(Note For another method,see Comparisan Test)
Ex Show that the series
n
diverges to
Sol
S
n
n(n )( n )
lim
S
S
diverges to
The given series diverges to
Ex Show that the series n diverges to
Sol
S
n ( n)
n(n )
lim
S
S
diverges to
The given series diverges to
Ex Test for convergence of the series
(i) ?( )
n
(ii) ? sin.
n
/
Sol
(i) Here ?( )
n
S
, S
, S
S
, S
,S
etc
S
, , , , , , ,which is not bounded
S
is not convergent
?( )
n is not convergent
(ii) ? sin
n
v
v
v
v
v
v
S
v
, S
v
v
v ,
S
v
v
v , S
v
v
v
v
etc
S
v
,v ,v ,
v
, , ,
v
,v
Clearly,lim supS
v
and lim inf S
It follows that S
is not convergent
Hence,the given series is not convergent
Ex Prove that the series ?
converges to
Sol
We have
S
For more notes, call 8130648819
[ .
/
]
4 S
a( r
)
r
5
S
[
]
lim
[
]
The sequence S
converges to
?u
converges to
Hence,the given series converges to
Article The geometric series x x
x
to
(i) converges if x i e |x| and converges to
( )
(ii) diverges if x
Proof (i) When |x|
Since |x| x
as n
Now
S
x x
x
to n terms
( x
)
x
x
x
x
lim
S
x
lim
4
x
x
5
lim
S
x
0 lim
r
,if |r| 1
the sequence S
is convergent
the given series is convergent
(ii) When x
Sub case I When x
S
to n terms n
lim
S
the sequence S
diverges to
the given series diverges to
Sub case II When x , x
as n
S
x x
to n terms
(x
)
x
lim
S
lim
4
x
x
5
x
lim
S
the sequence S
diverges to
the given series diverges to
Illustrations
?
n
converge , p -
?
n
diverge , p -
?
vn
v
v
diverge [ p
]
?
n
is convergent [ p
]
Note A geometric series converges only when absolute value of its common ratio is numerically less than
(b) Consider the series generated by (( )
)
that is,the series
( ) ?( )
( ) ( ) ( )
It is easily seen (by mathematical induction) that s
if n is even and s
if n is odd therefore,the
sequence of partial sums is ( , , , , ) Since this sequence is not convergent,the series ( ) is divergent
Ex Examine the convergence of the series
(i)
to (ii)
to
For more notes, call 8130648819
(iii) a b a
b
a
b
to (iv)
to
Sol
(i)
to
(
to ) (
to ) ?u
?v
(say)
now ?u
is a Geometric Series with common ratio
?u
is convergent
?v
is also a Geometric Series with common ratio
?v
is convergent
The given series ?(u
v
)is convergent ( The sum of two cgt series is also cgt )
(ii) Please try yourself
(iii) a b a
b
a
b
to
(a a
a
to ) (b b
b
to ) ?u
?v
(say)
Now ?u
is a G P with common ratio a and converges only when |a|
Similarly,?v
converges only when |b|
The given series ?(u
v
) converges only when both |a| and |b|
and in all other cases ?(u
v
) is divergent
(iv)
to
(
)
(
)
(
)
to
to
?
n
which is of the form ?
n
with p
y p series test,the series is convergent
The n
Term Test If a series ?u
is convergent,then lim
u
Is the converse true
Proof Let S denote the nth partial sum of the series ?u
Let ?u
is convergent S
is convergent
lim
S
is finite and unique
Let lim
S
s(say)
lim
S
s
Now S
u
u
u
u
S
u
u
u
S
S
u
lim
u
lim
(S
S
) lim
S
lim
S
s s
Hence ?u
is convergent lim
u
The converse of the above theorem is not always true
i e lim
u
but the series is not convergent
For example,the series
n
diverges,though lim
u
lim
n
Corollary If lim
u
,then the series ?u
cannot converge
Cauchy Criterion for Series The series ?x
converges if and only if for every there exists
M( ) such that if m n M( ),then
( ) |S
S
| |x
x
x
|
Proof The series x
is convergent iff the sequence S
of its partial sums is convergent
y Cauchy
s general principle of convergence for sequences
S
is convergent iff for each given ,there exists a positive integer m such that
|S
S
| m n
Page 5
For more notes, call 8130648819
INFINITE SERIES
Topics
1. Definition of infinite series, sequence of partial sums.
2. Cauchy criterion
3. Boundedness of sequence of partial sums,
4. Comparison test, limit comparison tests
5. D’ Alembert’s Ratio test, Cauchy’s nth root test
6. Raabe’s test, Logarithm test, Gauss test
7. Integral test (without proof)
8. Cauchy’s Condensation text
9. Alternating series, Leibniz test.
10. Absolute and Conditional convergence
In elementary texts,an infinite series is sometimes defined to be an expression of the form x
x
x
However,this definition lac s clarity,since there is a priori no particular value that we can attach to this array of
symbols,which call for an infinite number of additions to be performed
Def
If X (x
) is a sequence in ,then the infinite series (or simply the series) generated by X is the sequence
S (s
) defined by
s
x
s
s
x
( x
x
)
s
s
x
( x
x
x
)
The numbers x
are called the terms of the series and the numbers s
are called the partial sums of this series If limS
exits,(limit of sequence of partial sum) we say that this series is convergent and call this limit the sum or the value of
this series If this limit does not exists,we say that the series S is divergent
It is convenient to use symbols such as
( ) ?(x
) or ?x
or ? x
As a sequence may be indexed such that its first element is not x
,but is x
,or x
or x
,we will denote the series having
these numbers as their first element by the symbols
? x
or ? x
or ? x
Partial Sums
S
u
S
u
u
S
u
u
u
u
are called the first,second, ,nth partial sums of the series of ?u
Note To every infinite series u
there corresponds a sequence S
of its partial sums It should be noted that
when the first term in the series is x
,then the first partial sum is denoted by s
Warning The reader should guard against confusing the words sequence’’ and series’’ In nonmathematical language,
these words are interchangeable however,in mathematics these words are not synonyms Indeed,a series is a
sequence S (s
) obtained from a given sequence X (x
) according to the special procedure given in
Definition
series of positive terms
Def
If all the terms of the series u
u
u
u
are positive i e ,if u
n,then the
series u
is called a series of positive terms
Alternating series
Def
A series in which the terms are alternatively positive and negative is called an alternating series
Thus,the series ?( )
u
u
u
u
u
( )
u
where u
n is an alternating series
ehaviour of an Infinite Series
Def
An infinite series, u
converges,diverges or oscillates (finitely or infinitely) according as the sequence S
of its partial sums converges,diverges or oscillates (finitely or infinitely)
(i) The series u
converges (or is said to be convergent) if the sequence S
of its partial sums converges
Thus, u
is convergent if lim
S
Finite
(ii) The series u
diverges (or is said to be divergent) if the sequence S
of its partial sums diverges
Thus, u
is divergent if lim
S
or
(iii) The series u
oscillates finitely if the sequence S
of its partial sums oscillates finitely
Thus, u
oscillates finitely if S
is bounded and neither converges nor diverges
For more notes, call 8130648819
(iv) The series u
oscillates infitely if the sequence S
of its partial sums oscillates infinitely
Thus, u
oscillates infinitely if S
is unbounded and neither converges nor diverges
Ex Discuss the convergence of the series ?
n(n )
n(n )
to
Sol
Here u
n(n )
n
n
Putting n , , , ,n
u
, u
, u
, ,u
n
n
Now S
u
u
u
u
S
n
lim
S
S
converges to ?u
converges to
(Note For another method,see Comparisan Test)
Ex Show that the series
n
diverges to
Sol
S
n
n(n )( n )
lim
S
S
diverges to
The given series diverges to
Ex Show that the series n diverges to
Sol
S
n ( n)
n(n )
lim
S
S
diverges to
The given series diverges to
Ex Test for convergence of the series
(i) ?( )
n
(ii) ? sin.
n
/
Sol
(i) Here ?( )
n
S
, S
, S
S
, S
,S
etc
S
, , , , , , ,which is not bounded
S
is not convergent
?( )
n is not convergent
(ii) ? sin
n
v
v
v
v
v
v
S
v
, S
v
v
v ,
S
v
v
v , S
v
v
v
v
etc
S
v
,v ,v ,
v
, , ,
v
,v
Clearly,lim supS
v
and lim inf S
It follows that S
is not convergent
Hence,the given series is not convergent
Ex Prove that the series ?
converges to
Sol
We have
S
For more notes, call 8130648819
[ .
/
]
4 S
a( r
)
r
5
S
[
]
lim
[
]
The sequence S
converges to
?u
converges to
Hence,the given series converges to
Article The geometric series x x
x
to
(i) converges if x i e |x| and converges to
( )
(ii) diverges if x
Proof (i) When |x|
Since |x| x
as n
Now
S
x x
x
to n terms
( x
)
x
x
x
x
lim
S
x
lim
4
x
x
5
lim
S
x
0 lim
r
,if |r| 1
the sequence S
is convergent
the given series is convergent
(ii) When x
Sub case I When x
S
to n terms n
lim
S
the sequence S
diverges to
the given series diverges to
Sub case II When x , x
as n
S
x x
to n terms
(x
)
x
lim
S
lim
4
x
x
5
x
lim
S
the sequence S
diverges to
the given series diverges to
Illustrations
?
n
converge , p -
?
n
diverge , p -
?
vn
v
v
diverge [ p
]
?
n
is convergent [ p
]
Note A geometric series converges only when absolute value of its common ratio is numerically less than
(b) Consider the series generated by (( )
)
that is,the series
( ) ?( )
( ) ( ) ( )
It is easily seen (by mathematical induction) that s
if n is even and s
if n is odd therefore,the
sequence of partial sums is ( , , , , ) Since this sequence is not convergent,the series ( ) is divergent
Ex Examine the convergence of the series
(i)
to (ii)
to
For more notes, call 8130648819
(iii) a b a
b
a
b
to (iv)
to
Sol
(i)
to
(
to ) (
to ) ?u
?v
(say)
now ?u
is a Geometric Series with common ratio
?u
is convergent
?v
is also a Geometric Series with common ratio
?v
is convergent
The given series ?(u
v
)is convergent ( The sum of two cgt series is also cgt )
(ii) Please try yourself
(iii) a b a
b
a
b
to
(a a
a
to ) (b b
b
to ) ?u
?v
(say)
Now ?u
is a G P with common ratio a and converges only when |a|
Similarly,?v
converges only when |b|
The given series ?(u
v
) converges only when both |a| and |b|
and in all other cases ?(u
v
) is divergent
(iv)
to
(
)
(
)
(
)
to
to
?
n
which is of the form ?
n
with p
y p series test,the series is convergent
The n
Term Test If a series ?u
is convergent,then lim
u
Is the converse true
Proof Let S denote the nth partial sum of the series ?u
Let ?u
is convergent S
is convergent
lim
S
is finite and unique
Let lim
S
s(say)
lim
S
s
Now S
u
u
u
u
S
u
u
u
S
S
u
lim
u
lim
(S
S
) lim
S
lim
S
s s
Hence ?u
is convergent lim
u
The converse of the above theorem is not always true
i e lim
u
but the series is not convergent
For example,the series
n
diverges,though lim
u
lim
n
Corollary If lim
u
,then the series ?u
cannot converge
Cauchy Criterion for Series The series ?x
converges if and only if for every there exists
M( ) such that if m n M( ),then
( ) |S
S
| |x
x
x
|
Proof The series x
is convergent iff the sequence S
of its partial sums is convergent
y Cauchy
s general principle of convergence for sequences
S
is convergent iff for each given ,there exists a positive integer m such that
|S
S
| m n
For more notes, call 8130648819
|x
x
x
| m n
Hence the result
Prove with the help of Cauchy
s general principle of convergence that the Harmonic series
?
n
n
does not converge
Solution If possible,suppose the given series is convergent
Let
y Cauchy’s general principle of convergence,there exists a positive integer m such that
|
m
m
n
|
n m
m
m
n
n m (i)
y ta ing n m ,we see that
m
m
n
m
m
m
m
m
( m m
m
m
)
i e
m
m
n
where n m m
This contradicts ( )
Our supposition is wrong
The given series does not converge
Consider a positive term series ?u
,
We have S
u
u
u
u
S
u
or S
S
u
, n (next partial sum will be higher than previous because of all positive term)
S
S
and so the sequence <S
> of partial sums of ?u
is monotonically increasing
We now that a monotonically increasing sequence is convergent if and only if it is bounded above Hence we have the
following
Fundamental Test for a positive Term Series
Theorem Let (x
) be a sequence of non negative real numbers Then the series ?x
converges iff the
sequence S (s
) of partial sums is bounded In this case,
? x
lim(s
) sup*s
+
OR The necessary and sufficient condition for the convergence of a positive(zero also) term series ?u
is that the
sequence S
of its partial sums is bounded above
i e ?u
converges S
n and being some positive real number
Proof
(i) Suppose the sequence S
is bounded above Since the series ?u
is of positive terms
the sequence S
is monotonically increasing
Since every monotonically increasing sequence which is bounded above,converges,
S
is convergent u
converges
Conversely
Suppose ?u
converges
the sequence S
of its partial sums also converges
Every convergent sequence is bounded,
S
is bounded
In particular, S
is bounded above
Remark: We know that a monotonic sequence can either converge or diverge but cannot oscillate. Hence a positive
term series either converge or diverges.
Article A positive term series either converges or diverges to
Proof Let ?u
be a positive term series and S
be its n
partial sum
Then S
u
u
u
u
S
u
Read More
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FAQs on Infinite Series - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is an infinite series in mathematics? |  |
Ans. An infinite series in mathematics is the sum of an infinite sequence of numbers. It is represented as a sum of terms, where each term is obtained by adding the previous term by a common difference or ratio.
| 2. How is the sum of an infinite series calculated? | |
Ans. The sum of an infinite series can be calculated using various methods, such as the geometric series formula, telescoping series, or convergence tests like the ratio test or the integral test. These methods help determine if the series converges or diverges, and if it converges, they provide a way to calculate its sum.
| 3. What are some common types of infinite series? | |
Ans. Some common types of infinite series in mathematics include geometric series, arithmetic series, harmonic series, and power series. Each type has its own properties, convergence criteria, and formulas to calculate its sum.
| 4. How are infinite series used in real-life applications? | |
Ans. Infinite series have various applications in real-life, such as in physics, engineering, finance, and computer science. For example, they are used to model population growth, calculate compound interest, design electrical circuits, analyze signal processing, and solve differential equations.
| 5. What are some convergence tests used to determine the convergence of an infinite series? | |
Ans. Some common convergence tests used to determine the convergence of an infinite series are the ratio test, the root test, the integral test, the comparison test, and the alternating series test. These tests help analyze the behavior of the series and determine if it converges or diverges.
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