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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
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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.
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Additional Information about Logarithmic Test (Solved Exercise) for Mathematics Preparation
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