Infinite Series | Topic-wise Tests & Solved Examples for Mathematics PDF Download

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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
27 docs|150 tests

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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