Page 2
For more notes, call 8130648819
(ii) If
u
u
involves the number e,apply logarithmic test
Note
For application of Gauss Test,expand
u
u
in powers of
n
as
u
u
n
O (
n
)
where O(
n
) stands for terms of order
n
and higher powers of
n
Cauchy
s Integral Test
Statement If for x ,f(x) is a non negative monotonically decreasing integrable function of x such that
f(n) u
for all positive integral values of n,then the series ? u
and the improper integral? f(x)
dx converge
or diverge together
Ex Test for convergence of the series
(i) ?
n
(ii) ?
n
(iii) ?
n(n )
(iv) ?
vn
(v) ?
(n )
(vi) ?
n
/
Solution (i) Here
u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now ? f(x)
lim
? ( )
,
Consider,?
dx
x
tan
x|
tan
n tan
lim
tan
tan
(finite)
? f(x)
dx converges and,
Hence by Cauchy Integral Test,? u
is convergent
(ii) Here u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now,? f(x)
dx lim
?
dx
x
lim
6
log( x )|
7
,log log -
? f(x)
dx diverges
and hence by Integral Test, ? u
is divergent
(iii) Here
? f(x)
dx lim
6?
dx
x(x )
7 lim
? (
x
x
)
dx lim
,log log(x )|
-
lim
[log
|
]
log
log
log
log (finite)
Page 3
For more notes, call 8130648819
(ii) If
u
u
involves the number e,apply logarithmic test
Note
For application of Gauss Test,expand
u
u
in powers of
n
as
u
u
n
O (
n
)
where O(
n
) stands for terms of order
n
and higher powers of
n
Cauchy
s Integral Test
Statement If for x ,f(x) is a non negative monotonically decreasing integrable function of x such that
f(n) u
for all positive integral values of n,then the series ? u
and the improper integral? f(x)
dx converge
or diverge together
Ex Test for convergence of the series
(i) ?
n
(ii) ?
n
(iii) ?
n(n )
(iv) ?
vn
(v) ?
(n )
(vi) ?
n
/
Solution (i) Here
u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now ? f(x)
lim
? ( )
,
Consider,?
dx
x
tan
x|
tan
n tan
lim
tan
tan
(finite)
? f(x)
dx converges and,
Hence by Cauchy Integral Test,? u
is convergent
(ii) Here u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now,? f(x)
dx lim
?
dx
x
lim
6
log( x )|
7
,log log -
? f(x)
dx diverges
and hence by Integral Test, ? u
is divergent
(iii) Here
? f(x)
dx lim
6?
dx
x(x )
7 lim
? (
x
x
)
dx lim
,log log(x )|
-
lim
[log
|
]
log
log
log
log (finite)
For more notes, call 8130648819
? f(x)
dx converges
and hence,by integral Test,? u
is convergent
(iv) Try yourself ,Ans Divergent-
(v) Try yourself ,Ans Convergent-
(vi) Try yourself ,Ans Divergent-
Ex Using Integral test,show that the series ?
n
converges if p and diverges if p
Solution Here u
n
f(n)
f(x)
x
For x and p ,f(x) is non negative mono dereasing function
Cauchy Integral test is applicable
Case I When p
? f(x)
dx lim
6?
dx
x
7 lim
6? x
7 lim
x
p
|
Sub case (i) When p ,p is positive so that
? f(x)
dx lim
p
[
x
]
lim
p
[
n
] lim
p
[
n
]
? f(x)
dx
p
( )
p
finite
? f(x)
dx converges ? u
converges
Sub case (ii) When p , p is positive so that
? f(x)
dx lim
p
,x
-
? f(x)
dx lim
p
,n
-
p
, -
? f(x)
dx diverges ? u
diverges
Case II When p ,f(x)
x
? f(x)
dx lim
6?
x
dx7 lim
(logx)
log log
? f(x)
dx diverges
? u
diverges
Hence ? u
converges if p and diverges if, p
Ex Using integral test,discuss the convergence of the series ?
n(logn)
, p
Solution Here u
n(logn)
f(n)
f(x)
x(logx)
For x ,p ,f(x) is non negative and monotonically decreasing function,
y Cauchy
s integral test,
? u
and ? f(x)
dx converges or diverges together
Case I When p
Page 4
For more notes, call 8130648819
(ii) If
u
u
involves the number e,apply logarithmic test
Note
For application of Gauss Test,expand
u
u
in powers of
n
as
u
u
n
O (
n
)
where O(
n
) stands for terms of order
n
and higher powers of
n
Cauchy
s Integral Test
Statement If for x ,f(x) is a non negative monotonically decreasing integrable function of x such that
f(n) u
for all positive integral values of n,then the series ? u
and the improper integral? f(x)
dx converge
or diverge together
Ex Test for convergence of the series
(i) ?
n
(ii) ?
n
(iii) ?
n(n )
(iv) ?
vn
(v) ?
(n )
(vi) ?
n
/
Solution (i) Here
u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now ? f(x)
lim
? ( )
,
Consider,?
dx
x
tan
x|
tan
n tan
lim
tan
tan
(finite)
? f(x)
dx converges and,
Hence by Cauchy Integral Test,? u
is convergent
(ii) Here u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now,? f(x)
dx lim
?
dx
x
lim
6
log( x )|
7
,log log -
? f(x)
dx diverges
and hence by Integral Test, ? u
is divergent
(iii) Here
? f(x)
dx lim
6?
dx
x(x )
7 lim
? (
x
x
)
dx lim
,log log(x )|
-
lim
[log
|
]
log
log
log
log (finite)
For more notes, call 8130648819
? f(x)
dx converges
and hence,by integral Test,? u
is convergent
(iv) Try yourself ,Ans Divergent-
(v) Try yourself ,Ans Convergent-
(vi) Try yourself ,Ans Divergent-
Ex Using Integral test,show that the series ?
n
converges if p and diverges if p
Solution Here u
n
f(n)
f(x)
x
For x and p ,f(x) is non negative mono dereasing function
Cauchy Integral test is applicable
Case I When p
? f(x)
dx lim
6?
dx
x
7 lim
6? x
7 lim
x
p
|
Sub case (i) When p ,p is positive so that
? f(x)
dx lim
p
[
x
]
lim
p
[
n
] lim
p
[
n
]
? f(x)
dx
p
( )
p
finite
? f(x)
dx converges ? u
converges
Sub case (ii) When p , p is positive so that
? f(x)
dx lim
p
,x
-
? f(x)
dx lim
p
,n
-
p
, -
? f(x)
dx diverges ? u
diverges
Case II When p ,f(x)
x
? f(x)
dx lim
6?
x
dx7 lim
(logx)
log log
? f(x)
dx diverges
? u
diverges
Hence ? u
converges if p and diverges if, p
Ex Using integral test,discuss the convergence of the series ?
n(logn)
, p
Solution Here u
n(logn)
f(n)
f(x)
x(logx)
For x ,p ,f(x) is non negative and monotonically decreasing function,
y Cauchy
s integral test,
? u
and ? f(x)
dx converges or diverges together
Case I When p
For more notes, call 8130648819
I
? f(x)
dx lim
6?
dx
x(logx)
7 lim
6? (logx)
x
dx7 lim
6
(logx)
p
7
Sub case (i) When p ,p is positive so that
I
p
[
(logx)
]
p
[
(log )
(logn)
]
? f(x)dx
lim
I
lim
p
[
(log )
(logn)
]
p
[
(log )
]
(p )(log )
finite
? f(x)dx
converges ? u
converges
Sub case(ii) When p , p is positive so that
I
p
,(logp)
(log )
-
? f(x)dx
lim
I
lim
p
,(logn)
(log )
-
p
, (Log )
-
? f(x)dx
diverges
? u
diverges
Case II When p ,f(x)
x log x
I
?
dx
xlogx
?
x /
logx
dx
loglogx-
loglogn loglog
? f(x)dx
lim
I
lim
(loglogn loglog ) loglog
? f(x)dx
diverges
? u
diverges
? u
converges if p and diverges if p
Ex Discuss the convergence of the series ?
n log n (loglogn)
,
Solution u
n log (loglogn)
( ),n
f(x)
xlogx(loglogx)
,x
For x ,p f(x) is positive and decreasing
y Cauchy’s integral test,? u
and? f(x)dx
converges or diverges together
Case I When p
I
? f(x)dx
?
dx
xlogx (loglogx)
lim
[?(loglogx)
xlogx
dx]
lim
6
(loglogx)
p
7
7
Sub case (i) When p ,p is positive so that
Page 5
For more notes, call 8130648819
(ii) If
u
u
involves the number e,apply logarithmic test
Note
For application of Gauss Test,expand
u
u
in powers of
n
as
u
u
n
O (
n
)
where O(
n
) stands for terms of order
n
and higher powers of
n
Cauchy
s Integral Test
Statement If for x ,f(x) is a non negative monotonically decreasing integrable function of x such that
f(n) u
for all positive integral values of n,then the series ? u
and the improper integral? f(x)
dx converge
or diverge together
Ex Test for convergence of the series
(i) ?
n
(ii) ?
n
(iii) ?
n(n )
(iv) ?
vn
(v) ?
(n )
(vi) ?
n
/
Solution (i) Here
u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now ? f(x)
lim
? ( )
,
Consider,?
dx
x
tan
x|
tan
n tan
lim
tan
tan
(finite)
? f(x)
dx converges and,
Hence by Cauchy Integral Test,? u
is convergent
(ii) Here u
n
f(n)
f(x)
x
For x ,f(x) is non negative and a monotonically decreasing function of x
Cauchy
s integral test is applicable
Now,? f(x)
dx lim
?
dx
x
lim
6
log( x )|
7
,log log -
? f(x)
dx diverges
and hence by Integral Test, ? u
is divergent
(iii) Here
? f(x)
dx lim
6?
dx
x(x )
7 lim
? (
x
x
)
dx lim
,log log(x )|
-
lim
[log
|
]
log
log
log
log (finite)
For more notes, call 8130648819
? f(x)
dx converges
and hence,by integral Test,? u
is convergent
(iv) Try yourself ,Ans Divergent-
(v) Try yourself ,Ans Convergent-
(vi) Try yourself ,Ans Divergent-
Ex Using Integral test,show that the series ?
n
converges if p and diverges if p
Solution Here u
n
f(n)
f(x)
x
For x and p ,f(x) is non negative mono dereasing function
Cauchy Integral test is applicable
Case I When p
? f(x)
dx lim
6?
dx
x
7 lim
6? x
7 lim
x
p
|
Sub case (i) When p ,p is positive so that
? f(x)
dx lim
p
[
x
]
lim
p
[
n
] lim
p
[
n
]
? f(x)
dx
p
( )
p
finite
? f(x)
dx converges ? u
converges
Sub case (ii) When p , p is positive so that
? f(x)
dx lim
p
,x
-
? f(x)
dx lim
p
,n
-
p
, -
? f(x)
dx diverges ? u
diverges
Case II When p ,f(x)
x
? f(x)
dx lim
6?
x
dx7 lim
(logx)
log log
? f(x)
dx diverges
? u
diverges
Hence ? u
converges if p and diverges if, p
Ex Using integral test,discuss the convergence of the series ?
n(logn)
, p
Solution Here u
n(logn)
f(n)
f(x)
x(logx)
For x ,p ,f(x) is non negative and monotonically decreasing function,
y Cauchy
s integral test,
? u
and ? f(x)
dx converges or diverges together
Case I When p
For more notes, call 8130648819
I
? f(x)
dx lim
6?
dx
x(logx)
7 lim
6? (logx)
x
dx7 lim
6
(logx)
p
7
Sub case (i) When p ,p is positive so that
I
p
[
(logx)
]
p
[
(log )
(logn)
]
? f(x)dx
lim
I
lim
p
[
(log )
(logn)
]
p
[
(log )
]
(p )(log )
finite
? f(x)dx
converges ? u
converges
Sub case(ii) When p , p is positive so that
I
p
,(logp)
(log )
-
? f(x)dx
lim
I
lim
p
,(logn)
(log )
-
p
, (Log )
-
? f(x)dx
diverges
? u
diverges
Case II When p ,f(x)
x log x
I
?
dx
xlogx
?
x /
logx
dx
loglogx-
loglogn loglog
? f(x)dx
lim
I
lim
(loglogn loglog ) loglog
? f(x)dx
diverges
? u
diverges
? u
converges if p and diverges if p
Ex Discuss the convergence of the series ?
n log n (loglogn)
,
Solution u
n log (loglogn)
( ),n
f(x)
xlogx(loglogx)
,x
For x ,p f(x) is positive and decreasing
y Cauchy’s integral test,? u
and? f(x)dx
converges or diverges together
Case I When p
I
? f(x)dx
?
dx
xlogx (loglogx)
lim
[?(loglogx)
xlogx
dx]
lim
6
(loglogx)
p
7
7
Sub case (i) When p ,p is positive so that
For more notes, call 8130648819
I
p
[
(loglogx)
]
p
[
(loglog )
(loglogn)
]
? f(x)dx lim
I
lim
p
[
(loglog )
(loglogn)
]
p
[
(loglog )
]
(p )(loglog )
finite
? f(x)dx
converges
? u
converges
Sub Case (ii) When p , p is positive so that
I
p
,(loglogn)
(loglog )
-
? f(x)
dx lim
I
lim
p
,(loglogn)
(loglog )
-
p
, (loglog )
-
? f(x)
dx diverges ? u
diverges
Case II When p ,
f(x)
xlogx(loglogx)
I
?f(x)
dx ?
dx
xlog x(log logx)
?
xlogx
loglogx
dx
log(loglogx)-
log(loglogn) log(loglog )
? f(x)
dx lim
I
lim
,log(loglogn) log(loglog )- log(loglog )
? f(x)
dx diverges ? u
diverges,
Hence ? u
converges if p and diverges if p
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