Cauchy's Condensation Test (With Solved Exercise) Mathematics Notes | EduRev

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Mathematics : Cauchy's Condensation Test (With Solved Exercise) Mathematics Notes | EduRev

 Page 2


For more notes,  call 8130648819 
 
Since, logn is an increasing squence,  
 u
 
 is a decreasing sequence 
                                    u
 
 u
   
                n    
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together 
Now  
                             ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
log 
 
 ?
 
 
nlog 
 
   
 
 
log 
?
 
 
n
 
   
 
Consider v
 
 
 
 
n
            v
 
   
 
 
n
   
  
 lim
   
v
 
   
 lim
   
 
n
   
 
 
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
           ? 
 
u
 
 
 
   
 is divergent 
            ? u
 
 
 
   
is divergent  
(ii) Here  
                           u
 
 
 
nlogn
 
Since, nlogn is an increasing sequence, 
 u
 
 is a decreasing sequence  
i e               u
 
 u
   
                       n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
          ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
log 
 
 ?
 
nlog 
 
   
 
 
log 
?
 
n
 
   
 
Since,?
 
n
 
   
 is divergent   ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
(iii) Here  
              u
 
 
 
nvlogn
 
Since nvlogn is an increasing sequence,  u
 
 is a decreasing sequence 
          u
 
 u
   
                   n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
vlog 
 
 ?
 
vnlog 
 
   
 
 
vlog 
?
 
vn
 
   
 
Since,?
 
vn
 
   
 is divergent                     ( p 
 
 
  ) 
        ? 
 
u
 
 
 
   
 is divergent  
        ? u
 
 
   
 is divergent  
(iv) Do yourself  
(v) Here u
 
 
 
n(logn)
 
 
Page 3


For more notes,  call 8130648819 
 
Since, logn is an increasing squence,  
 u
 
 is a decreasing sequence 
                                    u
 
 u
   
                n    
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together 
Now  
                             ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
log 
 
 ?
 
 
nlog 
 
   
 
 
log 
?
 
 
n
 
   
 
Consider v
 
 
 
 
n
            v
 
   
 
 
n
   
  
 lim
   
v
 
   
 lim
   
 
n
   
 
 
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
           ? 
 
u
 
 
 
   
 is divergent 
            ? u
 
 
 
   
is divergent  
(ii) Here  
                           u
 
 
 
nlogn
 
Since, nlogn is an increasing sequence, 
 u
 
 is a decreasing sequence  
i e               u
 
 u
   
                       n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
          ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
log 
 
 ?
 
nlog 
 
   
 
 
log 
?
 
n
 
   
 
Since,?
 
n
 
   
 is divergent   ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
(iii) Here  
              u
 
 
 
nvlogn
 
Since nvlogn is an increasing sequence,  u
 
 is a decreasing sequence 
          u
 
 u
   
                   n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
vlog 
 
 ?
 
vnlog 
 
   
 
 
vlog 
?
 
vn
 
   
 
Since,?
 
vn
 
   
 is divergent                     ( p 
 
 
  ) 
        ? 
 
u
 
 
 
   
 is divergent  
        ? u
 
 
   
 is divergent  
(iv) Do yourself  
(v) Here u
 
 
 
n(logn)
 
 
For more notes,  call 8130648819 
 
Case I When p   
                              
 
n(logn)
 
 
 
n
                  n   
Since,?
 
n
 
   
 diverges,by comparison test ?
 
n(logn)
 
 
   
 diverges  
Case II When p    
Since n(logn)
 
 is an increasing sequence, u
 
 is a decreasing sequence  
i e                         u
 
 u
   
                    n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
           ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
(log 
 
 )
 
 ?
 
( log )
 
 
   
 
 
(log )
 
?
 
n
 
 
   
 
Since,?
 
n
 
 
   
 is convergent if p   and divergent if p    
         ? 
 
u
 
 
 
   
 is convergent if p   and divergent if p   
         ? u
 
 
   
 is convergent if p   and divergent if p     
Hence,? u
 
 
   
  is convergent if p   and divergent if p    
(vi) Here u
 
 
 
(logn)
 
 
Case I When p  ,u
 
   
Since, lim
   
u
 
    ,? 
 
 
   
 diverges  
Case II  when p  ,let p  q,where q    
Since, lim
   
u
 
 lim
   
 
(logn)
  
 lim
   
(logn)
 
     
        ? u
 
 
   
  diverges   
Case III When  p    
Since, (logn)
 
 is an increasing sequence, u
 
 is a decreasing sequence  
i e                         u
 
 u
    
                    n   
    y Cauchy
 
s  condensation  test,the series  ? u
 
 
   
 and ? 
 
 u
 
 
 
   
 converge or diverge together   
Now  ? 
 
 u
 
 
 
   
 ? 
 
  
 
   
 
 
(log 
 
)
 
 ? 
 
 
 
   
 
 
(nlog )
 
  
 
(log )
 
?
 
 
n
 
 
   
 
Consider, v
 
 
 
 
n
 
  so that v
 
  /
 
 
(n
  /
)
 
 
   lim
   
v
 
   
 lim
   
 
(n
 
 
)
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
         ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
Hence,? u
 
 
   
 is divergent for all values of p  
Page 4


For more notes,  call 8130648819 
 
Since, logn is an increasing squence,  
 u
 
 is a decreasing sequence 
                                    u
 
 u
   
                n    
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together 
Now  
                             ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
log 
 
 ?
 
 
nlog 
 
   
 
 
log 
?
 
 
n
 
   
 
Consider v
 
 
 
 
n
            v
 
   
 
 
n
   
  
 lim
   
v
 
   
 lim
   
 
n
   
 
 
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
           ? 
 
u
 
 
 
   
 is divergent 
            ? u
 
 
 
   
is divergent  
(ii) Here  
                           u
 
 
 
nlogn
 
Since, nlogn is an increasing sequence, 
 u
 
 is a decreasing sequence  
i e               u
 
 u
   
                       n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
          ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
log 
 
 ?
 
nlog 
 
   
 
 
log 
?
 
n
 
   
 
Since,?
 
n
 
   
 is divergent   ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
(iii) Here  
              u
 
 
 
nvlogn
 
Since nvlogn is an increasing sequence,  u
 
 is a decreasing sequence 
          u
 
 u
   
                   n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
vlog 
 
 ?
 
vnlog 
 
   
 
 
vlog 
?
 
vn
 
   
 
Since,?
 
vn
 
   
 is divergent                     ( p 
 
 
  ) 
        ? 
 
u
 
 
 
   
 is divergent  
        ? u
 
 
   
 is divergent  
(iv) Do yourself  
(v) Here u
 
 
 
n(logn)
 
 
For more notes,  call 8130648819 
 
Case I When p   
                              
 
n(logn)
 
 
 
n
                  n   
Since,?
 
n
 
   
 diverges,by comparison test ?
 
n(logn)
 
 
   
 diverges  
Case II When p    
Since n(logn)
 
 is an increasing sequence, u
 
 is a decreasing sequence  
i e                         u
 
 u
   
                    n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
           ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
(log 
 
 )
 
 ?
 
( log )
 
 
   
 
 
(log )
 
?
 
n
 
 
   
 
Since,?
 
n
 
 
   
 is convergent if p   and divergent if p    
         ? 
 
u
 
 
 
   
 is convergent if p   and divergent if p   
         ? u
 
 
   
 is convergent if p   and divergent if p     
Hence,? u
 
 
   
  is convergent if p   and divergent if p    
(vi) Here u
 
 
 
(logn)
 
 
Case I When p  ,u
 
   
Since, lim
   
u
 
    ,? 
 
 
   
 diverges  
Case II  when p  ,let p  q,where q    
Since, lim
   
u
 
 lim
   
 
(logn)
  
 lim
   
(logn)
 
     
        ? u
 
 
   
  diverges   
Case III When  p    
Since, (logn)
 
 is an increasing sequence, u
 
 is a decreasing sequence  
i e                         u
 
 u
    
                    n   
    y Cauchy
 
s  condensation  test,the series  ? u
 
 
   
 and ? 
 
 u
 
 
 
   
 converge or diverge together   
Now  ? 
 
 u
 
 
 
   
 ? 
 
  
 
   
 
 
(log 
 
)
 
 ? 
 
 
 
   
 
 
(nlog )
 
  
 
(log )
 
?
 
 
n
 
 
   
 
Consider, v
 
 
 
 
n
 
  so that v
 
  /
 
 
(n
  /
)
 
 
   lim
   
v
 
   
 lim
   
 
(n
 
 
)
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
         ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
Hence,? u
 
 
   
 is divergent for all values of p  
For more notes,  call 8130648819 
 
Ex    Using Cauchy
 
s condensation test,discuss the convergence of ?
logn
n
 
   
 
Solution Here 
                             u
 
 
logn
n
                        n 
Consider         f(x) 
logx
x
,x   
                          f
 
(x) 
x 
 
x
 logx
x
 
 
  logx
x
 
 
                          f
 
(x)   
                logx   
                      logx   
                             x e                                ( loge  ) 
 f(x) is a decreasing function when x e  
                          u
 
 u
   
                       (   e  ) 
  u
 
 is a decreasing sequence of positive terms  
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now,                ? 
 
u
 
 
 
   
 ? 
 
 
   
 
log 
 
 
 
 
                                                ? nlog 
 
   
 log ? n
 
   
 
Since,? n
 
   
 is divergent,? 
 
u
 
 
 
   
is divergent  
        ? u
 
 
   
 is divergent  
Ex    Using Cauchy
 
s condensation test,discuss the convergence of ? (
logn
n
)
 
 
   
 
Solution Here 
                             u
 
 (
logn
n
)
 
                                n 
consider         f(x) (
logx
x
)
 
,                                 x   
                         f
 
(x)  (
logx
x
) (
  logx
x
 
) 
                        f
 
(x)                   logx   
                                                                  logx          x e    ( loge  ) 
 f(x) is decreasing function where x e  
 u
 
 u
   
             n                               (   e  ) 
    u
 
 is a decreasing sequence of positive terms  
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
                    ? 
 
u
 
 
 
   
 ? 
 
 
   
 4
log 
 
 
 
5
 
 
                                         ? 
 
 
   
 
(nlog )
 
 
  
 (log )
 
?
n
 
 
 
 
   
 
Consider v
 
 
n
 
 
 
 so that v
 
   
 
 
 
    
  y Cauchy
 
s root test,? v
 
 
   
 converges  
 ? 
 
u
 
 
 
   
 converges ? u
 
 
   
 converges  
Page 5


For more notes,  call 8130648819 
 
Since, logn is an increasing squence,  
 u
 
 is a decreasing sequence 
                                    u
 
 u
   
                n    
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together 
Now  
                             ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
log 
 
 ?
 
 
nlog 
 
   
 
 
log 
?
 
 
n
 
   
 
Consider v
 
 
 
 
n
            v
 
   
 
 
n
   
  
 lim
   
v
 
   
 lim
   
 
n
   
 
 
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
           ? 
 
u
 
 
 
   
 is divergent 
            ? u
 
 
 
   
is divergent  
(ii) Here  
                           u
 
 
 
nlogn
 
Since, nlogn is an increasing sequence, 
 u
 
 is a decreasing sequence  
i e               u
 
 u
   
                       n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
          ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
log 
 
 ?
 
nlog 
 
   
 
 
log 
?
 
n
 
   
 
Since,?
 
n
 
   
 is divergent   ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
(iii) Here  
              u
 
 
 
nvlogn
 
Since nvlogn is an increasing sequence,  u
 
 is a decreasing sequence 
          u
 
 u
   
                   n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
vlog 
 
 ?
 
vnlog 
 
   
 
 
vlog 
?
 
vn
 
   
 
Since,?
 
vn
 
   
 is divergent                     ( p 
 
 
  ) 
        ? 
 
u
 
 
 
   
 is divergent  
        ? u
 
 
   
 is divergent  
(iv) Do yourself  
(v) Here u
 
 
 
n(logn)
 
 
For more notes,  call 8130648819 
 
Case I When p   
                              
 
n(logn)
 
 
 
n
                  n   
Since,?
 
n
 
   
 diverges,by comparison test ?
 
n(logn)
 
 
   
 diverges  
Case II When p    
Since n(logn)
 
 is an increasing sequence, u
 
 is a decreasing sequence  
i e                         u
 
 u
   
                    n   
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
           ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
 
 
(log 
 
 )
 
 ?
 
( log )
 
 
   
 
 
(log )
 
?
 
n
 
 
   
 
Since,?
 
n
 
 
   
 is convergent if p   and divergent if p    
         ? 
 
u
 
 
 
   
 is convergent if p   and divergent if p   
         ? u
 
 
   
 is convergent if p   and divergent if p     
Hence,? u
 
 
   
  is convergent if p   and divergent if p    
(vi) Here u
 
 
 
(logn)
 
 
Case I When p  ,u
 
   
Since, lim
   
u
 
    ,? 
 
 
   
 diverges  
Case II  when p  ,let p  q,where q    
Since, lim
   
u
 
 lim
   
 
(logn)
  
 lim
   
(logn)
 
     
        ? u
 
 
   
  diverges   
Case III When  p    
Since, (logn)
 
 is an increasing sequence, u
 
 is a decreasing sequence  
i e                         u
 
 u
    
                    n   
    y Cauchy
 
s  condensation  test,the series  ? u
 
 
   
 and ? 
 
 u
 
 
 
   
 converge or diverge together   
Now  ? 
 
 u
 
 
 
   
 ? 
 
  
 
   
 
 
(log 
 
)
 
 ? 
 
 
 
   
 
 
(nlog )
 
  
 
(log )
 
?
 
 
n
 
 
   
 
Consider, v
 
 
 
 
n
 
  so that v
 
  /
 
 
(n
  /
)
 
 
   lim
   
v
 
   
 lim
   
 
(n
 
 
)
 
      
  y Cauchy
 
s root test,?v
 
 is divergent  
         ? 
 
u
 
 
 
   
 is divergent 
         ? u
 
 
   
 is divergent  
Hence,? u
 
 
   
 is divergent for all values of p  
For more notes,  call 8130648819 
 
Ex    Using Cauchy
 
s condensation test,discuss the convergence of ?
logn
n
 
   
 
Solution Here 
                             u
 
 
logn
n
                        n 
Consider         f(x) 
logx
x
,x   
                          f
 
(x) 
x 
 
x
 logx
x
 
 
  logx
x
 
 
                          f
 
(x)   
                logx   
                      logx   
                             x e                                ( loge  ) 
 f(x) is a decreasing function when x e  
                          u
 
 u
   
                       (   e  ) 
  u
 
 is a decreasing sequence of positive terms  
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now,                ? 
 
u
 
 
 
   
 ? 
 
 
   
 
log 
 
 
 
 
                                                ? nlog 
 
   
 log ? n
 
   
 
Since,? n
 
   
 is divergent,? 
 
u
 
 
 
   
is divergent  
        ? u
 
 
   
 is divergent  
Ex    Using Cauchy
 
s condensation test,discuss the convergence of ? (
logn
n
)
 
 
   
 
Solution Here 
                             u
 
 (
logn
n
)
 
                                n 
consider         f(x) (
logx
x
)
 
,                                 x   
                         f
 
(x)  (
logx
x
) (
  logx
x
 
) 
                        f
 
(x)                   logx   
                                                                  logx          x e    ( loge  ) 
 f(x) is decreasing function where x e  
 u
 
 u
   
             n                               (   e  ) 
    u
 
 is a decreasing sequence of positive terms  
  y Cauchy
 
s condensation test,the series ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together  
Now 
                    ? 
 
u
 
 
 
   
 ? 
 
 
   
 4
log 
 
 
 
5
 
 
                                         ? 
 
 
   
 
(nlog )
 
 
  
 (log )
 
?
n
 
 
 
 
   
 
Consider v
 
 
n
 
 
 
 so that v
 
   
 
 
 
    
  y Cauchy
 
s root test,? v
 
 
   
 converges  
 ? 
 
u
 
 
 
   
 converges ? u
 
 
   
 converges  
For more notes,  call 8130648819 
 
Ex     Using Cauchy
 
s  condensation test,discuss the convergence of  ?
 
(nlogn)
 
 
   
 
Solution  Here  u
 
 
 
(nlogn)
 
,       n   
Case I  When  p  ,u
 
   
Since, lim
   
u
 
    , ? u
 
 
   
 is divergent   
Case II When p  ,let p  q  where q     
u
 
  
 
(nlogn)
  
 (nlogn)
 
  
                               u
 
   as n   
Since, lim
   
u
 
  , u
 
 is divergent   
Case III When p  , u
 
   is a decreasing sequence  
  y Cauchy
 
s condensation test,the series  ? u
 
 
   
 and ? 
 
u
 
 
 
   
 converge or diverge together   
Now       ? 
 
u
 
 
 
   
 ? 
 
 
   
 
 
( 
 
 log 
 
)
 
 
 ? 
 
 
   
 
 
( 
 
 nlog )
 
 ?
 
 
 (   )
 n
 
(log )
 
 
 
(log )
 
 
   
?
 
 
 (   )
n
 
 
   
 
If p  ,then  
 (   )
    so that 
 
 
 (   )
   and 
 
 
 (   )
 n
 
 
 
n
 
  
 ut  the series ?
 
n
 
 is convergent for p          
 ?
 
 
 (   )
 n
 
 
   
 is convegent   ? 
 
u
 
 
 
   
  is convergent for  p   
If  p  ,then  ? 
 
u
 
 
 
   
 
 
log 
?
 
n
 
   
 
 ut ?
 
n
 is divergent   
 ? 
 
u
 
 
 
   
 is divergent for p    
If p  ,then    p    so that  
 (   )
  ,i e 
 
 
 (   )
   and  
 
 
 (   )
   and 
 
 
 (   )
 n
 
 
 
n
 
 
 ut  the series ?
 
n
 
 is divergent for p     
Hence,? u
 
 
   
  is convergent for p   and divergent for p    
Def A series with terms alternatively positive and negative is called an alternating series  
Thus the series 
u
 
 u
 
 u
 
 u
 
            where u
 
       n  
is an alternating series and is briefly written as ?(  )
   
u
 
 
 
   
 
Leibnitz
 
s Test on Alternating Series  
Statement   The alternating series   
?(  )
    
u
 
 u
 
 u
 
 u
 
 u
 
               (u
 
     n) converges if  
(i) u
 
 u
   
   n and (ii) lim
   
u
 
   
Proof Let S
 
 denote the nth partial sum of the series   ?(  )
    
u
 
 
S
  
 u
 
 u
 
 u
 
 u
 
 u
 
   u
    
 u
    
 u
  
 
 u
 
 (u
 
 u
 
) (u
 
 u
 
)  (u
    
 u
    
) u
  
 
 u
 
 ,(u
 
 u
 
) (u
 
 u
 
)   (u
    
 u
    
) u
  
- 
 u
 
                      ,   u
 
 u
   
 and u
 
     n- 
 the sequence S
  
 is bounded above   
Also      S
    
 S
  
 u
    
 u
    
 
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