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Direct comparison Test and Limit Comparison Test
If an and bn are the nth terms of two infinite series, and if we knowthe behavior of one of them, then the behavior of the other can beobtained using the direct comparison test, provided the followingconditions are satisfied.
If 0 < an bn for all values of n, then
1 1
1. ,n n
n n
If b converges then a converges
1 1
2. ,n n
n n
If a diverges then b diverges
Direct Comparison Test
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Use direct comparison test to determine theconvergence or divergence of the series
an is the given series.
21
1
3 1n n
We have to identify a series bn whose behavior weknow and such that an < bn
Since the degree of the denominator of an
is 2, bn
alsomust be of degree 2.
The obvious choice is2
1n
bn
2 21 1
3 1n n
21
1
is a converging -seriesn pn
2 21 1
1 1converges when compared to
3 1n nn n
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Use direct comparison test to determine theconvergence or divergence of the series
What is a suitable bn to compare an?
The obvious choice is4
1n
b
n
4 4
1 1
3 1n n
1/ 41
1is a -series with p = 1/4
n
pn
1/441 1
1 1diverges when compared to
3 1n n nn
4
1
3 1na
n
1/ 4
1n
bn
41
1
3 1n n
Since p< 1, this is a diverging p-series.
bn < an and bn diverges
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Use direct comparison test to determine theconvergence or divergence of the series
What is a suitable bn to compare ansuch that an < bn
The obvious choice is2
1n
bn
21 1
for 3
!
n
n n
21
1is a converging -series with p >1
n
pn
21 1
1 1converges when compared to!n nn n
1
!na
n
0
1
!n n
0 1
1 11 also converges
! !n nn n
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If an and bn are the nth terms of two infinite series, and if we know
the behavior of one of them, then the behavior of the other can be
obtained using the limit comparison test.
If an > 0 and bn > 0 for all values of n and
n
n
n
aLim L
b
Limit Comparison Test
where L is finite and positive:
Review:1
What is ?n
Limn
As n increases, the denominatorgrows without bounds.
01
nLim
n
22What is ?
( 1)nn
Limn n
The numerator and the denominatorhave the same degree
The limit is the ratio ofthe leading coefficient.
22 2
(2
1) 1nn
Limn n
n nThe two series a and b with both converge or diverge.
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Use the limit comparison test to determine theconvergence or divergence of the series
What is a suitable bn whose behavior we know.
The obvious choice is1
nb
n
n
nn
aLim
b
2
1
1na
n
22
1
1n n
2
1
1
1n
nLim
n
2 1n
nLim
n
1
is a diverging harmonic
series.
This limit is finiteand positive.
nb
n na and b either both converge or diverge.
nn
Since b diverge a divergs, .es
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Use the limit comparison test to determine theconvergence or divergence of the series
What is a suitable bn whose behavior we know.
The obvious choice is1
nb
n
n
nn
aLim
b
2
5 3
2 5n
na
n n
2
5 3
2 51n
n
n nLim
n
2
(5 3)
2 5n
n nLim
n n
5
is a diverging harmonic
series.
This limit is finite and positive
22
5 3
2 5n
n
n n
nb
n na and b either both converge or diverge.
nn
Since b diverge a divergs, .es
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Use the limit comparison test to determine theconvergence or divergence of the series
What is a suitable bn whose behavior we know.
The obvious choice is1
2n n
b
n
nn
aLim
b
1( 1)2n
n
na
n
1( 1)2
1
2
n
n
n
n
nLim
1
2
2( 1)
n
n n
nLim
n
2
12 ( 1)2nn
n
n
2
( 1)n
nLim
n
This is a converging geometric series.
This limit is finite and positive
n na and b either both converge or diverge.
nn
Since b converge a convergs, .es
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Use the limit comparison test to determine theconvergence or divergence of the series
What is a suitable bn whose behavior we know.
The obvious choice is1
nb
n
n
nn
aLim
b
2
1
1na
n n
2
1
11n
n nLim
n
2 1n
nLim
n n
1
2
This is a diverging harmonic series.
21
1
1n n n
2n
nLim
n
This limit is finite and positive
n na and b either both converge or diverge.
nn
Since b diverge a divergs, .es
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1. nth Term Test
Review of Tests so far
2. Geometric Series Test
3. Telescopic Series Test
4. Integral Test
5. Direct Comparison Test
6. Limit Comparison Test
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Test the convergence or divergence of theseries: of the series
Which of the tests is appropriate in this case?
This is a geometric series with a = 5 and1
5r
1
5r
1
5 1
This is a converging geometric series.
Can you find the sum of this converging
geometric series?
0
15
5
n
n
Answer:25
6
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Use the limit comparison test to determine theconvergence or divergence of the series
What is a suitable bn whose behavior we know.
The obvious choice is2
1n
b
n
n
nn
aLim
b
2
1
3 2 15na n n
2
2
1
3 2 151n
n nLim
n
2
23 2 15n
nLim
n n
1
3
is a converging p-series.
This limit is finite and positive.
24
1
3 2 15n n n
Use limit comparison test
nb
n na and b either both converge or diverge.
nn
Since b converge a convergs, .es
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Test the convergence or divergence of theseries:
What is an appropriate test in this case?
When you are in doubt, always try the nth term test?
2 3n
nLim n
1
2
Since the limiting value of the nth term is not zero, this seriesdiverges.
1 2 3n
n
n
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Test the convergence or divergence of theseries: of the series
What is an appropriate test in this case?
Since the corresponding function can beintegrated, we should try integral test.
2 2( )
( 1)
xf x
x
2 21 ( 1)n
n
n
The corresponding function is:
2 21( )x
xdx
x2 + 1 = u
2xdx = du
xdx = du/2
2
1
2
u
u
d
1
2u 2
1
2( 1)x
2 21 ( 1)
xdxx
2 21
lim( 1)
b
bx dx
x 21
1lim2( 1)
b
b x
2
1 1
2( 1) 2(1 1)b
1
4 Since the integral converges,
the series converges.
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Test the convergence or divergence of theseries: of the series
What is an appropriate test in this case?
Use partial fractions to simplify
3
( 3)n n
3
A B
n n
Equate the numeratorand denominator.
3 = A(n + 3) + Bn Setting n = 0 gives:
( 3)
( 3)
A n bn
n n
A = 1
1
3
( 3)n n n
Setting n = -3 gives: B = -1
1
3( 3)n n n
1
1 1
3n n n
114
1 12 5
1 13 6
1 1 ......4 7
1 11
2 3
This is a converging
telescopic series.
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Given that:
2
2 21 3
1 1,
(2 1) 8 (2 1)n nfind
n n
2
21
1 1 101
(2 1) 9 9n n
21
1
(2 1)n n
2
2 21 3
1 1
(2 1) (2 1)n n
2
2 2 23 1 1
1 1 1
(2 1) (2 1) (2 1)n nn n n
2 10
8 9