Mathematics Exam > Mathematics Notes > Topic-wise Tests & Solved Examples for Mathematics > D'Morgan and Bertand's Test (Solved Exercise)
D'Morgan and Bertand's Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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
Read More
|
27 docs|150 tests
|
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.
Related Exams
About this Document
|
4.67/5 Rating |
|
Nov 08, 2024 Last updated |
Document Description: D'Morgan and Bertand's Test (Solved Exercise) for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation.
The notes and questions for D'Morgan and Bertand's Test (Solved Exercise) have been prepared according to the Mathematics exam syllabus. Information about D'Morgan and Bertand's Test (Solved Exercise) covers topics
like and D'Morgan and Bertand's Test (Solved Exercise) Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for D'Morgan and Bertand's Test (Solved Exercise).
Introduction of D'Morgan and Bertand's Test (Solved Exercise) in English is available as part of our Topic-wise Tests & Solved Examples for Mathematics
for Mathematics & D'Morgan and Bertand's Test (Solved Exercise) in Hindi for Topic-wise Tests & Solved Examples for Mathematics course.
Download more important topics related with notes, lectures and mock test series for Mathematics
Exam by signing up for free. Mathematics: D'Morgan and Bertand's Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
Description
Full syllabus notes, lecture & questions for D'Morgan and Bertand's Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics - Mathematics | Plus excerises question with solution to help you revise complete syllabus for Topic-wise Tests & Solved Examples for Mathematics | Best notes, free PDF download
Information about D'Morgan and Bertand's Test (Solved Exercise)
In this doc you can find the meaning of D'Morgan and Bertand's Test (Solved Exercise) defined & explained in the simplest way possible. Besides explaining types of
D'Morgan and Bertand's Test (Solved Exercise) theory, EduRev gives you an ample number of questions to practice D'Morgan and Bertand's Test (Solved Exercise) tests, examples and also practice Mathematics
tests
|
27 docs|150 tests
|
Download as PDF
|
Explore Courses for Mathematics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Extra Questions
,ppt
,Free
,Previous Year Questions with Solutions
,Sample Paper
,Exam
,D'Morgan and Bertand's Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,practice quizzes
,Viva Questions
,video lectures
,Objective type Questions
,D'Morgan and Bertand's Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,D'Morgan and Bertand's Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,Important questions
,past year papers
,study material
,MCQs
,Semester Notes
,Summary
,shortcuts and tricks
,mock tests for examination
;
Additional Information about D'Morgan and Bertand's Test (Solved Exercise) for Mathematics Preparation
D'Morgan and Bertand's Test (Solved Exercise) Free PDF Download
The D'Morgan and Bertand's Test (Solved Exercise) is an invaluable resource that delves deep into the core of the Mathematics exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the D'Morgan and Bertand's Test (Solved Exercise) now and kickstart your journey towards success in the Mathematics exam.
Importance of D'Morgan and Bertand's Test (Solved Exercise)
The importance of D'Morgan and Bertand's Test (Solved Exercise) cannot be overstated, especially for Mathematics aspirants.
This document holds the key to success in the Mathematics exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
D'Morgan and Bertand's Test (Solved Exercise) Notes
D'Morgan and Bertand's Test (Solved Exercise) Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to D'Morgan and Bertand's Test (Solved Exercise).
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, D'Morgan and Bertand's Test (Solved Exercise) Notes on EduRev are your ultimate resource for success.
D'Morgan and Bertand's Test (Solved Exercise) Mathematics Questions
The "D'Morgan and Bertand's Test (Solved Exercise) Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the
Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study D'Morgan and Bertand's Test (Solved Exercise) on the App
Students of Mathematics can study D'Morgan and Bertand's Test (Solved Exercise) alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the D'Morgan and Bertand's Test (Solved Exercise),
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of D'Morgan and Bertand's Test (Solved Exercise) is prepared as per the latest Mathematics syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup