D’Alembert's Ratio Test (With Solved Exercise) Mathematics Notes | EduRev

Topic-wise Tests & Solved Examples for IIT JAM Mathematics

Mathematics : D’Alembert's Ratio Test (With Solved Exercise) Mathematics Notes | EduRev

 Page 1


  
 
       
     
 
    
 
    
 
    
     
 
    
 
     
    
 
    
 
 
     
  
 
 
 
 
   
    
          
         
  
       
 
 
    
     
 
  
Article D’Alembert
 
s Ratio Test  
Statement If ?u
 
 is a series of positive terms such that  
(a)  lim
   
u
 
u
   
   then 
(i)     ?u
 
 is convergent if                            (ii) ?u
 
 is divergent if     
(iii)  ?u
 
 may converge or diverge if     (i e ,the test fails if    ) 
(b)   lim
   
u
 
u
   
  ,then the series ?u
 
 is convergent   
Proof (a) ?u
 
 is series of positive terms  
    u
 
                         n 
     
u
 
u
   
                      lim
   
u
 
u
   
     
Since    lim
   
u
 
u
   
  ,therefore,for each       a positive integer m such that  
|
u
 
u
   
  |               n  m 
                 
u
 
u
   
     
Replacing n by m,m  ,m  ,  ,n    in the above inequality,we have  
    
u
 
u
   
     
    
u
    
u
   
     
    
u
   
u
   
     
        
        
    
u
   
u
 
     
Multiplying the above (n m) inequality,we have  
(   )
   
 
u
 
u
 
 (   )
   
                              ( ) 
(i) Let     
Choose     such that      (This is always possible since we have only to choose an   such that 
         
From ( ),we have 
(   )
   
 
u
 
u
 
 
        u
 
 
u
 
(   )
   
 
        u
 
 u
 
(   )
 
 
 
(   )
 
 
          u
 
 K 
 
(   )
 
   n m           where K u
 
(   )
 
  
Now,?
 
(   )
 
 
   
 being a geometric series with common ratio 
 
   
   is convergent Therefore,  
Page 2


  
 
       
     
 
    
 
    
 
    
     
 
    
 
     
    
 
    
 
 
     
  
 
 
 
 
   
    
          
         
  
       
 
 
    
     
 
  
Article D’Alembert
 
s Ratio Test  
Statement If ?u
 
 is a series of positive terms such that  
(a)  lim
   
u
 
u
   
   then 
(i)     ?u
 
 is convergent if                            (ii) ?u
 
 is divergent if     
(iii)  ?u
 
 may converge or diverge if     (i e ,the test fails if    ) 
(b)   lim
   
u
 
u
   
  ,then the series ?u
 
 is convergent   
Proof (a) ?u
 
 is series of positive terms  
    u
 
                         n 
     
u
 
u
   
                      lim
   
u
 
u
   
     
Since    lim
   
u
 
u
   
  ,therefore,for each       a positive integer m such that  
|
u
 
u
   
  |               n  m 
                 
u
 
u
   
     
Replacing n by m,m  ,m  ,  ,n    in the above inequality,we have  
    
u
 
u
   
     
    
u
    
u
   
     
    
u
   
u
   
     
        
        
    
u
   
u
 
     
Multiplying the above (n m) inequality,we have  
(   )
   
 
u
 
u
 
 (   )
   
                              ( ) 
(i) Let     
Choose     such that      (This is always possible since we have only to choose an   such that 
         
From ( ),we have 
(   )
   
 
u
 
u
 
 
        u
 
 
u
 
(   )
   
 
        u
 
 u
 
(   )
 
 
 
(   )
 
 
          u
 
 K 
 
(   )
 
   n m           where K u
 
(   )
 
  
Now,?
 
(   )
 
 
   
 being a geometric series with common ratio 
 
   
   is convergent Therefore,  
For more notes,  call 8130648819 
 
by comparison test,the series ? u
 
 
   
 is convergent     
(ii) Let     
Choose     such that       (This is always possiblle since we have only to choose an   such that 
        
Since    ,        
From ( ),we have  
                  
u
 
u
 
 (   )
   
 
               u
 
 
u
 
(   )
   
 
               u
 
 u
 
(   )
 
 
 
(   )
 
 
               u
 
   
 
(   )
 
 n m where   u
 
(   )
 
 
Now,?
 
(   )
 
 
   
being a geometric series with common ratio
 
   
   is divergent Therefore,  
by comparison test,the series ? u
 
 
 
   
is divergent   
(iii) Let     
First consider the series  
              ?
 
n
   
 
 
 
 
 
   
 
n
   
              u
 
 
 
n
,                              u
   
 
 
n  
 
             
u
 
u
   
 
n  
n
   
 
n
  
so that  
              lim
   
u
 
u
   
   
Since the series ?
 
n
is divergent,we find that if    ,a series may diverge   
Next,consider the series  
               ?
 
n
 
 
 
 
 
 
 
 
 
   
 
n
 
   
u
 
 
 
n
 
,                          u
   
 
 
(n  )
 
 
              
u
 
u
   
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
so that Lim
   
u
 
u
   
   
Since the series ?
 
n
 
is convergent,we find that if   
  ,a series may converge The above two examples show that if    ,a series may converge or diverge  
Hence the test fails when      
Remar  Another equivalent form of Ratio Test is as follow  
If ?u
 
 is a positive term series such that lim
   
u
   
u
 
   
then 
(i) ?u
 
 is convergent if     
(ii) ?u
 
 is divergent if     
 
Ex    Discuss the convergence of the following series  
(i)   
  
 
 
 
  
 
 
 
  
 
 
                                 (ii)   
 
 
  
 
 
 
  
 
 
 
  
  ,(p  ) 
(iii)
 
 
 
 
 
 
 
 
   
 
 
   
  
             (iv)
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
   
Solution (i) Here u
 
 
n 
n
 
 
u
   
 
(n  ) 
(n  )
   
 
Page 3


  
 
       
     
 
    
 
    
 
    
     
 
    
 
     
    
 
    
 
 
     
  
 
 
 
 
   
    
          
         
  
       
 
 
    
     
 
  
Article D’Alembert
 
s Ratio Test  
Statement If ?u
 
 is a series of positive terms such that  
(a)  lim
   
u
 
u
   
   then 
(i)     ?u
 
 is convergent if                            (ii) ?u
 
 is divergent if     
(iii)  ?u
 
 may converge or diverge if     (i e ,the test fails if    ) 
(b)   lim
   
u
 
u
   
  ,then the series ?u
 
 is convergent   
Proof (a) ?u
 
 is series of positive terms  
    u
 
                         n 
     
u
 
u
   
                      lim
   
u
 
u
   
     
Since    lim
   
u
 
u
   
  ,therefore,for each       a positive integer m such that  
|
u
 
u
   
  |               n  m 
                 
u
 
u
   
     
Replacing n by m,m  ,m  ,  ,n    in the above inequality,we have  
    
u
 
u
   
     
    
u
    
u
   
     
    
u
   
u
   
     
        
        
    
u
   
u
 
     
Multiplying the above (n m) inequality,we have  
(   )
   
 
u
 
u
 
 (   )
   
                              ( ) 
(i) Let     
Choose     such that      (This is always possible since we have only to choose an   such that 
         
From ( ),we have 
(   )
   
 
u
 
u
 
 
        u
 
 
u
 
(   )
   
 
        u
 
 u
 
(   )
 
 
 
(   )
 
 
          u
 
 K 
 
(   )
 
   n m           where K u
 
(   )
 
  
Now,?
 
(   )
 
 
   
 being a geometric series with common ratio 
 
   
   is convergent Therefore,  
For more notes,  call 8130648819 
 
by comparison test,the series ? u
 
 
   
 is convergent     
(ii) Let     
Choose     such that       (This is always possiblle since we have only to choose an   such that 
        
Since    ,        
From ( ),we have  
                  
u
 
u
 
 (   )
   
 
               u
 
 
u
 
(   )
   
 
               u
 
 u
 
(   )
 
 
 
(   )
 
 
               u
 
   
 
(   )
 
 n m where   u
 
(   )
 
 
Now,?
 
(   )
 
 
   
being a geometric series with common ratio
 
   
   is divergent Therefore,  
by comparison test,the series ? u
 
 
 
   
is divergent   
(iii) Let     
First consider the series  
              ?
 
n
   
 
 
 
 
 
   
 
n
   
              u
 
 
 
n
,                              u
   
 
 
n  
 
             
u
 
u
   
 
n  
n
   
 
n
  
so that  
              lim
   
u
 
u
   
   
Since the series ?
 
n
is divergent,we find that if    ,a series may diverge   
Next,consider the series  
               ?
 
n
 
 
 
 
 
 
 
 
 
   
 
n
 
   
u
 
 
 
n
 
,                          u
   
 
 
(n  )
 
 
              
u
 
u
   
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
so that Lim
   
u
 
u
   
   
Since the series ?
 
n
 
is convergent,we find that if   
  ,a series may converge The above two examples show that if    ,a series may converge or diverge  
Hence the test fails when      
Remar  Another equivalent form of Ratio Test is as follow  
If ?u
 
 is a positive term series such that lim
   
u
   
u
 
   
then 
(i) ?u
 
 is convergent if     
(ii) ?u
 
 is divergent if     
 
Ex    Discuss the convergence of the following series  
(i)   
  
 
 
 
  
 
 
 
  
 
 
                                 (ii)   
 
 
  
 
 
 
  
 
 
 
  
  ,(p  ) 
(iii)
 
 
 
 
 
 
 
 
   
 
 
   
  
             (iv)
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
   
Solution (i) Here u
 
 
n 
n
 
 
u
   
 
(n  ) 
(n  )
   
 
For more notes,  call 8130648819 
 
u
 
u
   
 
n 
n
 
 
(n  )
   
(n  ) 
 
n 
n
 
 
(n  )
   
(n  )n 
  
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
 lim
   
u
 
u
   
 lim
   
(  
 
n
)
 
 e                            ,   e  - 
  y D’Alembert’s Ratio Test, u
 
 is convergent   
(ii) Do yourself                                                       ,Ans Convergent- 
(iii) Do yourself                                                      ,Ans Convergent- 
(iv) Do yourself                                                      ,Ans Convergent- 
JAM      MCQ Que     
Let S be the series ?
 
(    ) 
(    )
 
   
 and T be the series ? 4
    
    
5
(   )
 
 
   
 of real numbers   
Then,which one of the following is true  
(a)  oth the series S and T are convergent 
(b) S is convergent and T is divergent 
(c) S is divergent and T is convergent 
(d)  oth the series S and T are divergent 
Sol
 
 S ?
 
(    ) 
(    )
 
   
 ?
 
  
 
 
   
   convergent 
or Consider the u
 
 for S 
u
 
 
 
( n  ) 
(    )
 
u
   
 
 
, (n  )  - 
, (   )  -
 
 
, n  -, 
    
-
 
u
 
u
   
 
( n  )( 
    
)
( n  )( 
    
)
 
n.  
 
n
/( 
  
 )
n.  
 
n
/( 
  
 
  
)
 
 .  
 
n
/
.  
 
n
/
 
lim
   
u
 
u
   
     
by ratio test S is convergent 
Consider the u
 
 for T 
u
 
 (
 n  
 n  
)
(   )
 
 
u
   
 (
 n  
 n  
)
(   )
 
 
u
 
u
   
 (
 n  
 n  
)
   
 
 (
 n  
 n  
)
   
 
 :
  
 
n
  
 
n
;
   
 
 :
  
 
n
  
 
n
;
   
 
 
lim
   
u
 
u
   
 tends to   
So, T is divergent 
Ex    Test the convergence of the following series  
(i)
 
 
 
  
 
 
 
 
 
   
 
 
 
 
 
   
                  (ii)
 
 
 
 
 
 
  
  
 
  
  
 
   
   
   
(iii)
  
 
 
  
 
 
 
  
 
 
                                          (iv)
 
 
 
  
 
 
 
  
 
 
   
(v)
 
 
 
  
 
 
  
  
 
  
   
   
Solution (i) Here u
 
 
 
 
(n)
 
  
                        u
   
 
 
   
(n  )
 
  
 
                   
u
 
u
   
 
 
 
 
   
 
(n  )
 
  
n
 
  
   
 
 
 
n
 
  n  
n
 
  
 
 
 
 
  
 
n
 
 
n
 
  
 
n
 
 
Page 4


  
 
       
     
 
    
 
    
 
    
     
 
    
 
     
    
 
    
 
 
     
  
 
 
 
 
   
    
          
         
  
       
 
 
    
     
 
  
Article D’Alembert
 
s Ratio Test  
Statement If ?u
 
 is a series of positive terms such that  
(a)  lim
   
u
 
u
   
   then 
(i)     ?u
 
 is convergent if                            (ii) ?u
 
 is divergent if     
(iii)  ?u
 
 may converge or diverge if     (i e ,the test fails if    ) 
(b)   lim
   
u
 
u
   
  ,then the series ?u
 
 is convergent   
Proof (a) ?u
 
 is series of positive terms  
    u
 
                         n 
     
u
 
u
   
                      lim
   
u
 
u
   
     
Since    lim
   
u
 
u
   
  ,therefore,for each       a positive integer m such that  
|
u
 
u
   
  |               n  m 
                 
u
 
u
   
     
Replacing n by m,m  ,m  ,  ,n    in the above inequality,we have  
    
u
 
u
   
     
    
u
    
u
   
     
    
u
   
u
   
     
        
        
    
u
   
u
 
     
Multiplying the above (n m) inequality,we have  
(   )
   
 
u
 
u
 
 (   )
   
                              ( ) 
(i) Let     
Choose     such that      (This is always possible since we have only to choose an   such that 
         
From ( ),we have 
(   )
   
 
u
 
u
 
 
        u
 
 
u
 
(   )
   
 
        u
 
 u
 
(   )
 
 
 
(   )
 
 
          u
 
 K 
 
(   )
 
   n m           where K u
 
(   )
 
  
Now,?
 
(   )
 
 
   
 being a geometric series with common ratio 
 
   
   is convergent Therefore,  
For more notes,  call 8130648819 
 
by comparison test,the series ? u
 
 
   
 is convergent     
(ii) Let     
Choose     such that       (This is always possiblle since we have only to choose an   such that 
        
Since    ,        
From ( ),we have  
                  
u
 
u
 
 (   )
   
 
               u
 
 
u
 
(   )
   
 
               u
 
 u
 
(   )
 
 
 
(   )
 
 
               u
 
   
 
(   )
 
 n m where   u
 
(   )
 
 
Now,?
 
(   )
 
 
   
being a geometric series with common ratio
 
   
   is divergent Therefore,  
by comparison test,the series ? u
 
 
 
   
is divergent   
(iii) Let     
First consider the series  
              ?
 
n
   
 
 
 
 
 
   
 
n
   
              u
 
 
 
n
,                              u
   
 
 
n  
 
             
u
 
u
   
 
n  
n
   
 
n
  
so that  
              lim
   
u
 
u
   
   
Since the series ?
 
n
is divergent,we find that if    ,a series may diverge   
Next,consider the series  
               ?
 
n
 
 
 
 
 
 
 
 
 
   
 
n
 
   
u
 
 
 
n
 
,                          u
   
 
 
(n  )
 
 
              
u
 
u
   
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
so that Lim
   
u
 
u
   
   
Since the series ?
 
n
 
is convergent,we find that if   
  ,a series may converge The above two examples show that if    ,a series may converge or diverge  
Hence the test fails when      
Remar  Another equivalent form of Ratio Test is as follow  
If ?u
 
 is a positive term series such that lim
   
u
   
u
 
   
then 
(i) ?u
 
 is convergent if     
(ii) ?u
 
 is divergent if     
 
Ex    Discuss the convergence of the following series  
(i)   
  
 
 
 
  
 
 
 
  
 
 
                                 (ii)   
 
 
  
 
 
 
  
 
 
 
  
  ,(p  ) 
(iii)
 
 
 
 
 
 
 
 
   
 
 
   
  
             (iv)
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
   
Solution (i) Here u
 
 
n 
n
 
 
u
   
 
(n  ) 
(n  )
   
 
For more notes,  call 8130648819 
 
u
 
u
   
 
n 
n
 
 
(n  )
   
(n  ) 
 
n 
n
 
 
(n  )
   
(n  )n 
  
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
 lim
   
u
 
u
   
 lim
   
(  
 
n
)
 
 e                            ,   e  - 
  y D’Alembert’s Ratio Test, u
 
 is convergent   
(ii) Do yourself                                                       ,Ans Convergent- 
(iii) Do yourself                                                      ,Ans Convergent- 
(iv) Do yourself                                                      ,Ans Convergent- 
JAM      MCQ Que     
Let S be the series ?
 
(    ) 
(    )
 
   
 and T be the series ? 4
    
    
5
(   )
 
 
   
 of real numbers   
Then,which one of the following is true  
(a)  oth the series S and T are convergent 
(b) S is convergent and T is divergent 
(c) S is divergent and T is convergent 
(d)  oth the series S and T are divergent 
Sol
 
 S ?
 
(    ) 
(    )
 
   
 ?
 
  
 
 
   
   convergent 
or Consider the u
 
 for S 
u
 
 
 
( n  ) 
(    )
 
u
   
 
 
, (n  )  - 
, (   )  -
 
 
, n  -, 
    
-
 
u
 
u
   
 
( n  )( 
    
)
( n  )( 
    
)
 
n.  
 
n
/( 
  
 )
n.  
 
n
/( 
  
 
  
)
 
 .  
 
n
/
.  
 
n
/
 
lim
   
u
 
u
   
     
by ratio test S is convergent 
Consider the u
 
 for T 
u
 
 (
 n  
 n  
)
(   )
 
 
u
   
 (
 n  
 n  
)
(   )
 
 
u
 
u
   
 (
 n  
 n  
)
   
 
 (
 n  
 n  
)
   
 
 :
  
 
n
  
 
n
;
   
 
 :
  
 
n
  
 
n
;
   
 
 
lim
   
u
 
u
   
 tends to   
So, T is divergent 
Ex    Test the convergence of the following series  
(i)
 
 
 
  
 
 
 
 
 
   
 
 
 
 
 
   
                  (ii)
 
 
 
 
 
 
  
  
 
  
  
 
   
   
   
(iii)
  
 
 
  
 
 
 
  
 
 
                                          (iv)
 
 
 
  
 
 
 
  
 
 
   
(v)
 
 
 
  
 
 
  
  
 
  
   
   
Solution (i) Here u
 
 
 
 
(n)
 
  
                        u
   
 
 
   
(n  )
 
  
 
                   
u
 
u
   
 
 
 
 
   
 
(n  )
 
  
n
 
  
   
 
 
 
n
 
  n  
n
 
  
 
 
 
 
  
 
n
 
 
n
 
  
 
n
 
 
For more notes,  call 8130648819 
 
lim
   
u
 
u
   
 lim
   
 
 
 
  
 
n
 
 
n
 
  
 
n
 
 
 
 
   
  y D
 
Alembert
 
s Ratio Test,?u
 
is divergent   
(ii) The given series is  
       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
Here u
 
 
n
 
 
 
                                        u
   
 
(n  )
 
 
   
 
           
u
 
u
   
 
n
 
(n  )
 
 
 
   
 
 
   
n
 
n
 
.  
 
n
/
 
 
 
.  
 
n
/
 
 
            lim
   
u
 
u
   
 lim
   
 
.  
 
n
/
 
     
  y D
 
Alembert
 
s Ratio Test, u
 
 s convergent   
(iii) Do yourself                                                          ,Ans Divergent- 
(iv) u
 
 
n 
 
 
, u
   
 
(n  ) 
 
   
divergent  
       
u
 
u
    
 
n 
 
 
 
 
   
(n  ) 
 
 
n  
 
         lim
   
u
 
u
   
 
 
 
     
  y D
 
Alembert Ratio Test,the series is divergent   
(v) Do yourself                                               ,Ans Divergent- 
Ex    Examine the convergence or divergence of the following series  
(i)
 
 
 
   
   
 
     
     
 
        
       
                 (ii)
 
   
 
 
 
 
 
  
 
 
 
 
 
  
                     (iii) ?
 
   
 
 
  
 
Solution (i) Here 
                               u
 
 
       n
      ,  (n  )  -
 
       n
      ( n  )
  
                        u
   
 
       n(n  )
      ( n  )( n  )
 
                        
u
 
u
   
 
       n
      ( n  )
     ( n  )( n  )
       n(n  )
  
                          
u
 
u
    
 
( n  )
n  
 
  
 
n
  
 
n
  
lim
   
u
 
u
    
 lim
   
  
 
n
  
 
n
     
  y D
 
Alembert’s Ratio  Test, u
 
  is convergent   
(ii) Here u
 
 
 
 
 
 
  
 
          u
   
 
 
   
 
   
  
 
u
 
u
   
 
 
 
 
   
 
 
    
  
 
 
  
 
 
 
 
 
   
.  
 
 
   
/
 
 
.  
 
 
 
/
 
 
 
 
  
 
 
   
  
 
 
 
 
 lim
   
u
 
u
   
 lim
   
 
 
 
  
 
 
   
  
 
 
 
 
 
 
   
  y D
 
Alembert’s Ratio Test, u
 
 is divergent  
(iii) Do yourself                                                        ,Ans Convergent- 
Ex    Test the convergence of the following series  
(i) ?
 
n 
                              (ii) ?
n
 
 a
 
 
 a
                          (iii) ?
n
 
(n  )
 
n 
                      (iv) ?
 
 
 n 
n
 
 
Solution (i) Do yourself                                                                      ,Ans Convergent- 
Page 5


  
 
       
     
 
    
 
    
 
    
     
 
    
 
     
    
 
    
 
 
     
  
 
 
 
 
   
    
          
         
  
       
 
 
    
     
 
  
Article D’Alembert
 
s Ratio Test  
Statement If ?u
 
 is a series of positive terms such that  
(a)  lim
   
u
 
u
   
   then 
(i)     ?u
 
 is convergent if                            (ii) ?u
 
 is divergent if     
(iii)  ?u
 
 may converge or diverge if     (i e ,the test fails if    ) 
(b)   lim
   
u
 
u
   
  ,then the series ?u
 
 is convergent   
Proof (a) ?u
 
 is series of positive terms  
    u
 
                         n 
     
u
 
u
   
                      lim
   
u
 
u
   
     
Since    lim
   
u
 
u
   
  ,therefore,for each       a positive integer m such that  
|
u
 
u
   
  |               n  m 
                 
u
 
u
   
     
Replacing n by m,m  ,m  ,  ,n    in the above inequality,we have  
    
u
 
u
   
     
    
u
    
u
   
     
    
u
   
u
   
     
        
        
    
u
   
u
 
     
Multiplying the above (n m) inequality,we have  
(   )
   
 
u
 
u
 
 (   )
   
                              ( ) 
(i) Let     
Choose     such that      (This is always possible since we have only to choose an   such that 
         
From ( ),we have 
(   )
   
 
u
 
u
 
 
        u
 
 
u
 
(   )
   
 
        u
 
 u
 
(   )
 
 
 
(   )
 
 
          u
 
 K 
 
(   )
 
   n m           where K u
 
(   )
 
  
Now,?
 
(   )
 
 
   
 being a geometric series with common ratio 
 
   
   is convergent Therefore,  
For more notes,  call 8130648819 
 
by comparison test,the series ? u
 
 
   
 is convergent     
(ii) Let     
Choose     such that       (This is always possiblle since we have only to choose an   such that 
        
Since    ,        
From ( ),we have  
                  
u
 
u
 
 (   )
   
 
               u
 
 
u
 
(   )
   
 
               u
 
 u
 
(   )
 
 
 
(   )
 
 
               u
 
   
 
(   )
 
 n m where   u
 
(   )
 
 
Now,?
 
(   )
 
 
   
being a geometric series with common ratio
 
   
   is divergent Therefore,  
by comparison test,the series ? u
 
 
 
   
is divergent   
(iii) Let     
First consider the series  
              ?
 
n
   
 
 
 
 
 
   
 
n
   
              u
 
 
 
n
,                              u
   
 
 
n  
 
             
u
 
u
   
 
n  
n
   
 
n
  
so that  
              lim
   
u
 
u
   
   
Since the series ?
 
n
is divergent,we find that if    ,a series may diverge   
Next,consider the series  
               ?
 
n
 
 
 
 
 
 
 
 
 
   
 
n
 
   
u
 
 
 
n
 
,                          u
   
 
 
(n  )
 
 
              
u
 
u
   
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
so that Lim
   
u
 
u
   
   
Since the series ?
 
n
 
is convergent,we find that if   
  ,a series may converge The above two examples show that if    ,a series may converge or diverge  
Hence the test fails when      
Remar  Another equivalent form of Ratio Test is as follow  
If ?u
 
 is a positive term series such that lim
   
u
   
u
 
   
then 
(i) ?u
 
 is convergent if     
(ii) ?u
 
 is divergent if     
 
Ex    Discuss the convergence of the following series  
(i)   
  
 
 
 
  
 
 
 
  
 
 
                                 (ii)   
 
 
  
 
 
 
  
 
 
 
  
  ,(p  ) 
(iii)
 
 
 
 
 
 
 
 
   
 
 
   
  
             (iv)
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
 
 
 
  
 
  
   
Solution (i) Here u
 
 
n 
n
 
 
u
   
 
(n  ) 
(n  )
   
 
For more notes,  call 8130648819 
 
u
 
u
   
 
n 
n
 
 
(n  )
   
(n  ) 
 
n 
n
 
 
(n  )
   
(n  )n 
  
 
(n  )
 
n
 
 (
n  
n
)
 
 (  
 
n
)
 
 
 lim
   
u
 
u
   
 lim
   
(  
 
n
)
 
 e                            ,   e  - 
  y D’Alembert’s Ratio Test, u
 
 is convergent   
(ii) Do yourself                                                       ,Ans Convergent- 
(iii) Do yourself                                                      ,Ans Convergent- 
(iv) Do yourself                                                      ,Ans Convergent- 
JAM      MCQ Que     
Let S be the series ?
 
(    ) 
(    )
 
   
 and T be the series ? 4
    
    
5
(   )
 
 
   
 of real numbers   
Then,which one of the following is true  
(a)  oth the series S and T are convergent 
(b) S is convergent and T is divergent 
(c) S is divergent and T is convergent 
(d)  oth the series S and T are divergent 
Sol
 
 S ?
 
(    ) 
(    )
 
   
 ?
 
  
 
 
   
   convergent 
or Consider the u
 
 for S 
u
 
 
 
( n  ) 
(    )
 
u
   
 
 
, (n  )  - 
, (   )  -
 
 
, n  -, 
    
-
 
u
 
u
   
 
( n  )( 
    
)
( n  )( 
    
)
 
n.  
 
n
/( 
  
 )
n.  
 
n
/( 
  
 
  
)
 
 .  
 
n
/
.  
 
n
/
 
lim
   
u
 
u
   
     
by ratio test S is convergent 
Consider the u
 
 for T 
u
 
 (
 n  
 n  
)
(   )
 
 
u
   
 (
 n  
 n  
)
(   )
 
 
u
 
u
   
 (
 n  
 n  
)
   
 
 (
 n  
 n  
)
   
 
 :
  
 
n
  
 
n
;
   
 
 :
  
 
n
  
 
n
;
   
 
 
lim
   
u
 
u
   
 tends to   
So, T is divergent 
Ex    Test the convergence of the following series  
(i)
 
 
 
  
 
 
 
 
 
   
 
 
 
 
 
   
                  (ii)
 
 
 
 
 
 
  
  
 
  
  
 
   
   
   
(iii)
  
 
 
  
 
 
 
  
 
 
                                          (iv)
 
 
 
  
 
 
 
  
 
 
   
(v)
 
 
 
  
 
 
  
  
 
  
   
   
Solution (i) Here u
 
 
 
 
(n)
 
  
                        u
   
 
 
   
(n  )
 
  
 
                   
u
 
u
   
 
 
 
 
   
 
(n  )
 
  
n
 
  
   
 
 
 
n
 
  n  
n
 
  
 
 
 
 
  
 
n
 
 
n
 
  
 
n
 
 
For more notes,  call 8130648819 
 
lim
   
u
 
u
   
 lim
   
 
 
 
  
 
n
 
 
n
 
  
 
n
 
 
 
 
   
  y D
 
Alembert
 
s Ratio Test,?u
 
is divergent   
(ii) The given series is  
       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
Here u
 
 
n
 
 
 
                                        u
   
 
(n  )
 
 
   
 
           
u
 
u
   
 
n
 
(n  )
 
 
 
   
 
 
   
n
 
n
 
.  
 
n
/
 
 
 
.  
 
n
/
 
 
            lim
   
u
 
u
   
 lim
   
 
.  
 
n
/
 
     
  y D
 
Alembert
 
s Ratio Test, u
 
 s convergent   
(iii) Do yourself                                                          ,Ans Divergent- 
(iv) u
 
 
n 
 
 
, u
   
 
(n  ) 
 
   
divergent  
       
u
 
u
    
 
n 
 
 
 
 
   
(n  ) 
 
 
n  
 
         lim
   
u
 
u
   
 
 
 
     
  y D
 
Alembert Ratio Test,the series is divergent   
(v) Do yourself                                               ,Ans Divergent- 
Ex    Examine the convergence or divergence of the following series  
(i)
 
 
 
   
   
 
     
     
 
        
       
                 (ii)
 
   
 
 
 
 
 
  
 
 
 
 
 
  
                     (iii) ?
 
   
 
 
  
 
Solution (i) Here 
                               u
 
 
       n
      ,  (n  )  -
 
       n
      ( n  )
  
                        u
   
 
       n(n  )
      ( n  )( n  )
 
                        
u
 
u
   
 
       n
      ( n  )
     ( n  )( n  )
       n(n  )
  
                          
u
 
u
    
 
( n  )
n  
 
  
 
n
  
 
n
  
lim
   
u
 
u
    
 lim
   
  
 
n
  
 
n
     
  y D
 
Alembert’s Ratio  Test, u
 
  is convergent   
(ii) Here u
 
 
 
 
 
 
  
 
          u
   
 
 
   
 
   
  
 
u
 
u
   
 
 
 
 
   
 
 
    
  
 
 
  
 
 
 
 
 
   
.  
 
 
   
/
 
 
.  
 
 
 
/
 
 
 
 
  
 
 
   
  
 
 
 
 
 lim
   
u
 
u
   
 lim
   
 
 
 
  
 
 
   
  
 
 
 
 
 
 
   
  y D
 
Alembert’s Ratio Test, u
 
 is divergent  
(iii) Do yourself                                                        ,Ans Convergent- 
Ex    Test the convergence of the following series  
(i) ?
 
n 
                              (ii) ?
n
 
 a
 
 
 a
                          (iii) ?
n
 
(n  )
 
n 
                      (iv) ?
 
 
 n 
n
 
 
Solution (i) Do yourself                                                                      ,Ans Convergent- 
For more notes,  call 8130648819 
 
(ii) Here u
 
 
n
 
 a
 
 
 a
                u
   
 
(n  )
 
 a
 
   
 a
 
                        
u
 
u
   
 
(n
 
 a)( 
   
 a)
((n  )
 
 a)( 
 
 a)
 
                                   
n
 
.  
a
n
 
/
n
 
6.  
 
n
/
 
 
a
n
 
7
 
 
   
.  
a
 
   
/
 
 
.  
a
 
 
/
 
                                  
  
a
n
 
.  
 
n
/
 
 
a
n
 
 
 .  
a
 
   
/
  
a
 
 
 
              lim
   
u
 
u
   
 
   
   
   
   
   
     
  y D
 
Alembert
 
s Ratio Test, u
 
 is convergent   
(iii) Do yourself                                                                          ,Ans Convergent- 
(iv) Do yourself                                                                           ,Ans Convergent- 
Ex    Discuss the convergence or divergence of the following series  
(i) ?
n
 
n 
                 (ii) ?
n 
n
 
                    (iii) ?
 
 
 n 
n
 
 
Solution (i) Here u
 
 
n
 
n 
 
 u
   
 
(n  )
 
(n  ) 
 
(n  )
 
(n  )n 
 
n  
n 
 
 
u
 
u
   
 
n
 
n  
 
 
 
n
 
 
n
 
 
 lim
   
u
 
u
   
 lim
   
 
 
n
 
 
n
 
 
 
 
     
  y D
 
Alembert
 
s Ratio Test, u
 
 is convergent   
(ii) Do yourself                                                                        ,Ans Convergent- 
(iii) Do yourself                                                                       ,Ans Divergent- 
Ex    Discuss the convergence or divergence of the following series  
(i) ?
x
 
 
 
 n
 
,x                (ii) ?
x
 
a vn
                      (iii) ?
v
n  
n
 
  
 x
 
 
(iv) ?
vn
vn
 
  
x
 
               (v) ?
x
 
n
,x                     (vi) ?
n
n
 
  
x
 
,x   
Solution (i) Here u
 
 
x
 
 
 
 n
 
 
 u
   
 
x
   
 
   
 (n  )
 
 
 
u
 
u
   
 
x
 
x
   
 
 
   
 
 
 
(n  )
 
n
 
 
 
x
 (  
 
n
)
 
  
 lim
   
u
 
u
   
 lim
   
 
x
(  
 
n
)
 
 
 
x
 
  y D
 
Alembert
 
s Ratio Test, u
 
 converges if 
 
x
   i e x   and diverges if 
 
x
  i e ,x   
When x  , 
               lim
   
u
 
u
   
   
 Ratio Test Fails   
Now,when x  , 
          u
 
 
 
 
 
 
 n
 
 
 
n
 
 
        ?u
 
 ?
 
n
 
 is of the form ?
 
n
 
 with p     
          ?u
 
 is convergent   
Hence the given series  u
 
converges if x   and diverges if x    
(ii) Do yourself                               ,Ans converges if x   and diverges if x  - 
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