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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
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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.
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