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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