Mathematics Exam > Mathematics Notes > Topic-wise Tests & Solved Examples for Mathematics > Cauchy's Condensation Test (With Solved Exercise)
Cauchy's Condensation Test (With 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
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
Read More
|
27 docs|150 tests
|
FAQs on Cauchy's Condensation Test (With Solved Exercise) - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is Cauchy's Condensation Test? |  |
Ans. Cauchy's Condensation Test is a mathematical test used to determine the convergence or divergence of a series. It involves comparing the terms of the series with a condensed version of the series, where the terms are multiplied by a power of 2. If the condensed series converges, then the original series also converges, and vice versa.
| 2. How is Cauchy's Condensation Test applied to determine the convergence or divergence of a series? | |
Ans. To apply Cauchy's Condensation Test, we multiply the terms of the series by increasing powers of 2. We then compare the condensed series with the original series. If the condensed series converges, then the original series also converges. If the condensed series diverges, then the original series also diverges.
| 3. Can Cauchy's Condensation Test be used for all types of series? | |
Ans. No, Cauchy's Condensation Test is specifically designed for series with non-negative terms. It cannot be applied to series with negative terms or alternating series. For such series, other convergence tests such as the Alternating Series Test or the Ratio Test should be used.
| 4. Are there any limitations or exceptions to Cauchy's Condensation Test? | |
Ans. Cauchy's Condensation Test may not be conclusive for series with terms that do not decrease monotonically. If the terms of the series do not decrease or increase in a consistent manner, the test may not accurately determine the convergence or divergence of the series.
| 5. Can Cauchy's Condensation Test be used to determine the sum of a convergent series? | |
Ans. No, Cauchy's Condensation Test only determines the convergence or divergence of a series, but it does not provide information about the sum of a convergent series. To find the sum of a convergent series, other methods such as the Geometric Series Sum formula or the telescoping series method should be used.
Related Exams
About this Document
|
4.77/5 Rating |
|
Dec 18, 2024 Last updated |
Document Description: Cauchy's Condensation Test (With Solved Exercise) for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation.
The notes and questions for Cauchy's Condensation Test (With Solved Exercise) have been prepared according to the Mathematics exam syllabus. Information about Cauchy's Condensation Test (With Solved Exercise) covers topics
like and Cauchy's Condensation Test (With Solved Exercise) Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Cauchy's Condensation Test (With Solved Exercise).
Introduction of Cauchy's Condensation Test (With Solved Exercise) in English is available as part of our Topic-wise Tests & Solved Examples for Mathematics
for Mathematics & Cauchy's Condensation Test (With 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: Cauchy's Condensation Test (With Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
Description
Full syllabus notes, lecture & questions for Cauchy's Condensation Test (With 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 Cauchy's Condensation Test (With Solved Exercise)
In this doc you can find the meaning of Cauchy's Condensation Test (With Solved Exercise) defined & explained in the simplest way possible. Besides explaining types of
Cauchy's Condensation Test (With Solved Exercise) theory, EduRev gives you an ample number of questions to practice Cauchy's Condensation Test (With 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
Important questions
,past year papers
,Previous Year Questions with Solutions
,study material
,Cauchy's Condensation Test (With Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,MCQs
,Semester Notes
,Free
,Extra Questions
,video lectures
,Sample Paper
,shortcuts and tricks
,ppt
,mock tests for examination
,Viva Questions
,Exam
,Summary
,practice quizzes
,Cauchy's Condensation Test (With Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,Objective type Questions
,Cauchy's Condensation Test (With Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
;
Additional Information about Cauchy's Condensation Test (With Solved Exercise) for Mathematics Preparation
Cauchy's Condensation Test (With Solved Exercise) Free PDF Download
The Cauchy's Condensation Test (With 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 Cauchy's Condensation Test (With Solved Exercise) now and kickstart your journey towards success in the Mathematics exam.
Importance of Cauchy's Condensation Test (With Solved Exercise)
The importance of Cauchy's Condensation Test (With 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.
Cauchy's Condensation Test (With Solved Exercise) Notes
Cauchy's Condensation Test (With 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 Cauchy's Condensation Test (With 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, Cauchy's Condensation Test (With Solved Exercise) Notes on EduRev are your ultimate resource for success.
Cauchy's Condensation Test (With Solved Exercise) Mathematics Questions
The "Cauchy's Condensation Test (With 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 Cauchy's Condensation Test (With Solved Exercise) on the App
Students of Mathematics can study Cauchy's Condensation Test (With Solved Exercise) alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Cauchy's Condensation Test (With 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 Cauchy's Condensation Test (With 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