Mathematics Exam > Mathematics Notes > Topic-wise Tests & Solved Examples for Mathematics > Logarithmic Test (Solved Exercise)
Logarithmic 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
If x e,then the ratio test fails
Now,put x e,in equation ( )
u
u
.
n
/
.
n
/
e
Since,the expression
u
u
involves the number e,so we apply logarithmic test
log
u
u
(n )log(
n
) (n )log(
n
) log e
(n )[log(
n
) log(
n
)]
(n )[(
n
n
n
) (
n
n
n
)]
(n )[
n
n
]
n
n
n
n
n
lim
log
u
u
lim
n[
n
n
] lim
(
n
)
y logarithm test,the series diverges
Hence,the given series ?u
converges if x e and diverges if x e
Ex Discuss the convergence of the series x
x
x
x
x
Solution u
n
x
n
and u
(n )
(n )
x
u
u
n
x
n
(n )
(n )
x
n
n
(n )n
(n )
x
.
n
/
x
( )
lim
u
u
ex
4 lim
(
n
)
e5
y D
Alembert
s Ratio Test,the series
{
converges if
ex
,i e x
e
and
diverges if
ex
i e x
e
If x
e
,then Ratio test fails
Put x
e
,in equation ( )
u
u
.
n
/
e
Since,
u
u
involves the number e,we apply logarithm test
log
u
u
loge nlog(
n
)
n(
n
n
n
)
n
n
lim
[nlog
u
u
] lim
(
n
)
y Logarithm test,the series is divergent
Hence,the given series ?u
converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
x
x
x
x
Solution Here
u
n
x
n
Page 3
For more notes, call 8130648819
If x e,then the ratio test fails
Now,put x e,in equation ( )
u
u
.
n
/
.
n
/
e
Since,the expression
u
u
involves the number e,so we apply logarithmic test
log
u
u
(n )log(
n
) (n )log(
n
) log e
(n )[log(
n
) log(
n
)]
(n )[(
n
n
n
) (
n
n
n
)]
(n )[
n
n
]
n
n
n
n
n
lim
log
u
u
lim
n[
n
n
] lim
(
n
)
y logarithm test,the series diverges
Hence,the given series ?u
converges if x e and diverges if x e
Ex Discuss the convergence of the series x
x
x
x
x
Solution u
n
x
n
and u
(n )
(n )
x
u
u
n
x
n
(n )
(n )
x
n
n
(n )n
(n )
x
.
n
/
x
( )
lim
u
u
ex
4 lim
(
n
)
e5
y D
Alembert
s Ratio Test,the series
{
converges if
ex
,i e x
e
and
diverges if
ex
i e x
e
If x
e
,then Ratio test fails
Put x
e
,in equation ( )
u
u
.
n
/
e
Since,
u
u
involves the number e,we apply logarithm test
log
u
u
loge nlog(
n
)
n(
n
n
n
)
n
n
lim
[nlog
u
u
] lim
(
n
)
y Logarithm test,the series is divergent
Hence,the given series ?u
converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
x
x
x
x
Solution Here
u
n
x
n
For more notes, call 8130648819
and u
(n )
x
(n )
u
u
n
x
n
(n )
(n )
x
(n )n
(n )
x
n
.
n
/
n
.
n
/
x
.
n
/
.
n
/
x
lim
u
u
ex
y D
Alembert
s ratio test,the series converges if
ex
i e x
e
and diverges if
ex
i e x
e
if x
e
,the test fails
Now,put x
e
in equation ( )
u
u
.
n
/
e
Since,the expression for
u
u
involves the number e,we apply logarithm test
log
u
u
loge (n )log(
n
)
(n )[
n
n
n
]
n[
n
n
n
]
n
n
(
n
n
)
n
n
n
n
lim
nlog
u
u
lim
[
n
]
y logarithm test,the series converges
Hence,the given series converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
a x
(a x )
(a x )
Solution Here
u
(a nx)
n
and u
,a (n )x-
(n )
u
u
(n )
n
(a nx)
,a (n )x-
(n ) n
x
.
a
nx
/
(n )
x
[
a
(n )x
]
.
a
nx
/
.
n
/
[
a
(n )x
]
x
( )
lim
u
u
e
/
e e
/
x
ex
[ lim
.
p
n
/
e
]
y D
Alembert
s Ratio Test,the series conveges if
ex
i e x
e
and diverges if
ex
i e if x
e
If x
e
,the Ratio test fails
Put x
e
in ( ) we get
Page 4
For more notes, call 8130648819
If x e,then the ratio test fails
Now,put x e,in equation ( )
u
u
.
n
/
.
n
/
e
Since,the expression
u
u
involves the number e,so we apply logarithmic test
log
u
u
(n )log(
n
) (n )log(
n
) log e
(n )[log(
n
) log(
n
)]
(n )[(
n
n
n
) (
n
n
n
)]
(n )[
n
n
]
n
n
n
n
n
lim
log
u
u
lim
n[
n
n
] lim
(
n
)
y logarithm test,the series diverges
Hence,the given series ?u
converges if x e and diverges if x e
Ex Discuss the convergence of the series x
x
x
x
x
Solution u
n
x
n
and u
(n )
(n )
x
u
u
n
x
n
(n )
(n )
x
n
n
(n )n
(n )
x
.
n
/
x
( )
lim
u
u
ex
4 lim
(
n
)
e5
y D
Alembert
s Ratio Test,the series
{
converges if
ex
,i e x
e
and
diverges if
ex
i e x
e
If x
e
,then Ratio test fails
Put x
e
,in equation ( )
u
u
.
n
/
e
Since,
u
u
involves the number e,we apply logarithm test
log
u
u
loge nlog(
n
)
n(
n
n
n
)
n
n
lim
[nlog
u
u
] lim
(
n
)
y Logarithm test,the series is divergent
Hence,the given series ?u
converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
x
x
x
x
Solution Here
u
n
x
n
For more notes, call 8130648819
and u
(n )
x
(n )
u
u
n
x
n
(n )
(n )
x
(n )n
(n )
x
n
.
n
/
n
.
n
/
x
.
n
/
.
n
/
x
lim
u
u
ex
y D
Alembert
s ratio test,the series converges if
ex
i e x
e
and diverges if
ex
i e x
e
if x
e
,the test fails
Now,put x
e
in equation ( )
u
u
.
n
/
e
Since,the expression for
u
u
involves the number e,we apply logarithm test
log
u
u
loge (n )log(
n
)
(n )[
n
n
n
]
n[
n
n
n
]
n
n
(
n
n
)
n
n
n
n
lim
nlog
u
u
lim
[
n
]
y logarithm test,the series converges
Hence,the given series converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
a x
(a x )
(a x )
Solution Here
u
(a nx)
n
and u
,a (n )x-
(n )
u
u
(n )
n
(a nx)
,a (n )x-
(n ) n
x
.
a
nx
/
(n )
x
[
a
(n )x
]
.
a
nx
/
.
n
/
[
a
(n )x
]
x
( )
lim
u
u
e
/
e e
/
x
ex
[ lim
.
p
n
/
e
]
y D
Alembert
s Ratio Test,the series conveges if
ex
i e x
e
and diverges if
ex
i e if x
e
If x
e
,the Ratio test fails
Put x
e
in ( ) we get
For more notes, call 8130648819
u
u
e.
ae
n
/
.
ae
n
/
.
n
/
log
u
u
[loge nlog(
n
)] 0nlog.
ae
n
/ (n )log.
ae
n
/1
n(
n
n
n
)
6n4
ae
n
a
e
n
a
e
n
5 (n )8
ae
n
a
e
(n )
a
e
(n )
9 7
(
n
n
) 64
a
e
n
a
e
n
5 4
a
e
(n )
a
e
(n )
57
nlog
u
u
(
) 4
a
e
a
e
n
5
n
n
4
a
e
a
e
(n )
5
lim
[n log
u
u
]
a
e
a
e
y logtest,the series diverges
Hence,the given series converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
Solution Here
u
( n )
( n )
and u
( n )
( n )
( n )
( n )
u
u
( n )
( n )
n
.
n
/
n
.
n
/
.
n
/
.
n
/
lim
u
u
Now,the ratio test fails
n(
u
u
) n6
( n )
( n )
7
n6
n
n ( n
n )
( n )
7
n
( n )
( n )
n
n
( n )
n
.
n
/
lim
(
u
u
)
Raabe
s test also fails Now,we apply Gauss test
u
u
( n )
( n )
.
n
/
.
n
/
(
n
)
(
n
)
(
n
n
) (
n
n
)
n
n
(
)
n
n
n
O(
n
)
Comparing it with
u
u
n
O(
n
),
we have
Thus,by Gauss test,the series ?u
diverges
Que framing ta e n
term numerator as product of odd and denominator as product of even
Page 5
For more notes, call 8130648819
If x e,then the ratio test fails
Now,put x e,in equation ( )
u
u
.
n
/
.
n
/
e
Since,the expression
u
u
involves the number e,so we apply logarithmic test
log
u
u
(n )log(
n
) (n )log(
n
) log e
(n )[log(
n
) log(
n
)]
(n )[(
n
n
n
) (
n
n
n
)]
(n )[
n
n
]
n
n
n
n
n
lim
log
u
u
lim
n[
n
n
] lim
(
n
)
y logarithm test,the series diverges
Hence,the given series ?u
converges if x e and diverges if x e
Ex Discuss the convergence of the series x
x
x
x
x
Solution u
n
x
n
and u
(n )
(n )
x
u
u
n
x
n
(n )
(n )
x
n
n
(n )n
(n )
x
.
n
/
x
( )
lim
u
u
ex
4 lim
(
n
)
e5
y D
Alembert
s Ratio Test,the series
{
converges if
ex
,i e x
e
and
diverges if
ex
i e x
e
If x
e
,then Ratio test fails
Put x
e
,in equation ( )
u
u
.
n
/
e
Since,
u
u
involves the number e,we apply logarithm test
log
u
u
loge nlog(
n
)
n(
n
n
n
)
n
n
lim
[nlog
u
u
] lim
(
n
)
y Logarithm test,the series is divergent
Hence,the given series ?u
converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
x
x
x
x
Solution Here
u
n
x
n
For more notes, call 8130648819
and u
(n )
x
(n )
u
u
n
x
n
(n )
(n )
x
(n )n
(n )
x
n
.
n
/
n
.
n
/
x
.
n
/
.
n
/
x
lim
u
u
ex
y D
Alembert
s ratio test,the series converges if
ex
i e x
e
and diverges if
ex
i e x
e
if x
e
,the test fails
Now,put x
e
in equation ( )
u
u
.
n
/
e
Since,the expression for
u
u
involves the number e,we apply logarithm test
log
u
u
loge (n )log(
n
)
(n )[
n
n
n
]
n[
n
n
n
]
n
n
(
n
n
)
n
n
n
n
lim
nlog
u
u
lim
[
n
]
y logarithm test,the series converges
Hence,the given series converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
a x
(a x )
(a x )
Solution Here
u
(a nx)
n
and u
,a (n )x-
(n )
u
u
(n )
n
(a nx)
,a (n )x-
(n ) n
x
.
a
nx
/
(n )
x
[
a
(n )x
]
.
a
nx
/
.
n
/
[
a
(n )x
]
x
( )
lim
u
u
e
/
e e
/
x
ex
[ lim
.
p
n
/
e
]
y D
Alembert
s Ratio Test,the series conveges if
ex
i e x
e
and diverges if
ex
i e if x
e
If x
e
,the Ratio test fails
Put x
e
in ( ) we get
For more notes, call 8130648819
u
u
e.
ae
n
/
.
ae
n
/
.
n
/
log
u
u
[loge nlog(
n
)] 0nlog.
ae
n
/ (n )log.
ae
n
/1
n(
n
n
n
)
6n4
ae
n
a
e
n
a
e
n
5 (n )8
ae
n
a
e
(n )
a
e
(n )
9 7
(
n
n
) 64
a
e
n
a
e
n
5 4
a
e
(n )
a
e
(n )
57
nlog
u
u
(
) 4
a
e
a
e
n
5
n
n
4
a
e
a
e
(n )
5
lim
[n log
u
u
]
a
e
a
e
y logtest,the series diverges
Hence,the given series converges if x
e
and diverges if x
e
Ex Discuss the convergence of the series
Solution Here
u
( n )
( n )
and u
( n )
( n )
( n )
( n )
u
u
( n )
( n )
n
.
n
/
n
.
n
/
.
n
/
.
n
/
lim
u
u
Now,the ratio test fails
n(
u
u
) n6
( n )
( n )
7
n6
n
n ( n
n )
( n )
7
n
( n )
( n )
n
n
( n )
n
.
n
/
lim
(
u
u
)
Raabe
s test also fails Now,we apply Gauss test
u
u
( n )
( n )
.
n
/
.
n
/
(
n
)
(
n
)
(
n
n
) (
n
n
)
n
n
(
)
n
n
n
O(
n
)
Comparing it with
u
u
n
O(
n
),
we have
Thus,by Gauss test,the series ?u
diverges
Que framing ta e n
term numerator as product of odd and denominator as product of even
For more notes, call 8130648819
Ex Discuss the convergence of the series
Solution Neglecting the first term,we have
u
( n )
( n )
and u
( n )
( n )
( n )
( n )
u
u
( n )
( n )
.
n
/
.
n
/
lim
u
u
Ratio test fails
Now,
u
u
(
n
)
(
n
)
(
n
n
) (
n
n
) (on expanding by inomial Theorem)
n
n
n
n
n
n
n
n
O(
n
)
Comparing it with
u
u
n
O(
n
),
we have
y Gauss test, the series ?u
diverges
Ex Test for convergence of the positive term series
( )( )
( )( )
Solution Neglecting the first terms,we have
u
( )( ) (n )
( )( ) (n )
and u
( )( ) (n )(n )
( )( ) (n )(n )
u
u
(n )
(n )
n
n
lim
u
u
The Ratio test fails
Now,
u
u
(
n
) (
n
)
(
n
) (
n
)
n
O(
n
)
y Gauss test,the series is convergent if ,i e and divergent if ,i e if
Read More
|
27 docs|150 tests
|
FAQs on Logarithmic Test (Solved Exercise) - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is a logarithmic test? |  |
Ans. A logarithmic test is a type of assessment or exam that focuses on questions and problems related to logarithmic functions and their properties. It typically involves solving equations involving logarithms, manipulating logarithmic expressions, and understanding the various properties of logarithmic functions.
| 2. How do logarithmic tests evaluate mathematical skills? | |
Ans. Logarithmic tests evaluate a student's mathematical skills by assessing their understanding and proficiency in solving logarithmic equations, applying logarithmic properties, and manipulating logarithmic expressions. The test may also gauge the student's ability to analyze and interpret logarithmic functions in real-world contexts.
| 3. What topics are covered in a logarithmic test? | |
Ans. A logarithmic test usually covers a range of topics related to logarithmic functions, including rules of logarithms, logarithmic equations, exponential and logarithmic functions, logarithmic identities, logarithmic differentiation, and applications of logarithms. It may also include questions on logarithmic scales and their use in various fields.
| 4. How can I prepare for a logarithmic test effectively? | |
Ans. To prepare for a logarithmic test effectively, it is recommended to review the fundamental concepts of logarithms, practice solving logarithmic equations, understand the properties of logarithmic functions, and work on various applications of logarithms. Additionally, solving practice problems and seeking guidance from textbooks, online resources, or a tutor can be beneficial.
| 5. Can you provide any tips for solving logarithmic problems efficiently in a test? | |
Ans. Yes, here are some tips for solving logarithmic problems efficiently in a test:
- Familiarize yourself with the rules and properties of logarithms to simplify expressions and equations.
- Practice converting between logarithmic and exponential forms to better understand the relationship between the two.
- Use logarithmic identities to simplify complex expressions and equations.
- Pay attention to the given instructions and ensure that the final answer is in the required form.
- When solving logarithmic equations, check for extraneous solutions that may arise due to restrictions on the domain.
Related Exams
About this Document
|
2K Views |
|
4.95/5 Rating |
|
Dec 18, 2024 Last updated |
Document Description: Logarithmic Test (Solved Exercise) for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation.
The notes and questions for Logarithmic Test (Solved Exercise) have been prepared according to the Mathematics exam syllabus. Information about Logarithmic Test (Solved Exercise) covers topics
like and Logarithmic Test (Solved Exercise) Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Logarithmic Test (Solved Exercise).
Introduction of Logarithmic Test (Solved Exercise) in English is available as part of our Topic-wise Tests & Solved Examples for Mathematics
for Mathematics & Logarithmic 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: Logarithmic Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
Description
Full syllabus notes, lecture & questions for Logarithmic 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 Logarithmic Test (Solved Exercise)
In this doc you can find the meaning of Logarithmic Test (Solved Exercise) defined & explained in the simplest way possible. Besides explaining types of
Logarithmic Test (Solved Exercise) theory, EduRev gives you an ample number of questions to practice Logarithmic 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
Semester Notes
,Previous Year Questions with Solutions
,Viva Questions
,Objective type Questions
,shortcuts and tricks
,Important questions
,Exam
,video lectures
,Summary
,MCQs
,past year papers
,Logarithmic Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,Logarithmic Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
,practice quizzes
,Extra Questions
,mock tests for examination
,study material
,ppt
,Free
,Sample Paper
,Logarithmic Test (Solved Exercise) | Topic-wise Tests & Solved Examples for Mathematics
;
Additional Information about Logarithmic Test (Solved Exercise) for Mathematics Preparation
Logarithmic Test (Solved Exercise) Free PDF Download
The Logarithmic 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 Logarithmic Test (Solved Exercise) now and kickstart your journey towards success in the Mathematics exam.
Importance of Logarithmic Test (Solved Exercise)
The importance of Logarithmic 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.
Logarithmic Test (Solved Exercise) Notes
Logarithmic 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 Logarithmic 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, Logarithmic Test (Solved Exercise) Notes on EduRev are your ultimate resource for success.
Logarithmic Test (Solved Exercise) Mathematics Questions
The "Logarithmic 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 Logarithmic Test (Solved Exercise) on the App
Students of Mathematics can study Logarithmic Test (Solved Exercise) alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Logarithmic 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 Logarithmic 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