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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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27 docs|150 tests
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FAQs on D'Morgan and Bertand's Test (Solved Exercise) - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is D'Morgan and Bertrand's Test? |  |
Ans. D'Morgan and Bertrand's Test is a mathematical exam or assessment that evaluates a student's understanding and application of mathematical concepts, particularly in the area of logic and set theory. It assesses the ability to solve problems using D'Morgan's Laws and Bertrand's postulate.
| 2. How can I prepare for D'Morgan and Bertrand's Test? | |
Ans. To prepare for D'Morgan and Bertrand's Test, it is important to review and understand the concepts of logic and set theory, specifically D'Morgan's Laws and Bertrand's postulate. Practice solving problems that involve these concepts to improve your skills and familiarity with the material. Utilize study resources such as textbooks, online tutorials, and practice exams to reinforce your understanding.
| 3. What are D'Morgan's Laws? | |
Ans. D'Morgan's Laws are a set of logical equivalences named after the mathematician Augustus De Morgan. These laws describe the relationship between logical operators "and," "or," and "not" in terms of set theory. The two laws are as follows:
1) The negation of a conjunction is the disjunction of the negations.
2) The negation of a disjunction is the conjunction of the negations.
These laws are often used to simplify logical expressions and prove mathematical statements.
| 4. What is Bertrand's postulate? | |
Ans. Bertrand's postulate, proposed by Joseph Bertrand, states that for any integer greater than 1, there always exists at least one prime number between that integer and its double. This postulate helps establish the existence of prime numbers within a given range and has significant applications in number theory and prime number distribution.
| 5. How can I improve my problem-solving skills for D'Morgan and Bertrand's Test? | |
Ans. Improving problem-solving skills for D'Morgan and Bertrand's Test can be achieved through regular practice and exposure to various problem-solving scenarios. Engage in exercises and problem sets that involve logical reasoning and set theory. Break down complex problems into smaller, manageable steps and practice applying D'Morgan's Laws and Bertrand's postulate to solve them. Seek help from teachers, tutors, or online resources to clarify any doubts and learn alternative problem-solving strategies. Regularly challenging yourself with new and diverse problems will help sharpen your skills and improve your performance on the test.
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The content of D'Morgan and Bertand's Test (Solved Exercise) is prepared as per the latest Mathematics syllabus.
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