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Goal: Does a series converge or diverge?
Lecture 24 – Divergence Test
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?1
kka
Divergence Test (If a series converges, then sequence converges to 0.)
1 13
12
k k
kExample 1 – Converge/Diverge?
2
Example 2 – Converge/Diverge?
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1tank
k
0
1
3
2
kk
k
Example 3 – Converge/Diverge?
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0
1
3
2
kk
k
However,
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1
n n
Example 4 – Converge/Diverge?
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1
1
n n
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
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However,
Goal: Does a series converge or diverge?
then, allfor function decreasing
positive, ,continuous a with )( If
Nx
fnfan
Lecture 25 – Integral Test
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?1
12
n n
Integral Test (The area under a function and infinite sum of the terms in a sequence defined by that function are related.)
N a
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If area under curve is bounded,
then so is
But then is a bounded,
monotonic sequence.
So it converges and
thus converges.
1N a2N a
1N N 1N
N a
If area under curve is unbounded,
then
is also unbounded.
And thus, diverges.
1N a2N a
1N N 1N
12
1
n n
Example 1 – Converge/Diverge?
7
2
1lim
n
n
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19
1
1 2 3
0 sequence converges to zero.
No info from Divergence Test.
1
1
nne
Example 2 – Converge/Diverge?
8
nn e
1lim 0
n
n e1
1
sequence converges to zero.
No info from Divergence Test.
2
ln
n n
nExample 3 – Converge/Diverge?
9
n
n
n
lnlim 0 sequence converges to zero.
No info from Divergence Test.
2
2ln
3
3ln
4
4ln
1 2 3
021
1
k k
Example 4 – Converge/Diverge?
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21
1lim
n
n0 sequence converges to zero.
No info from Divergence Test.
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Lecture 26 – Ratio and Root Tests
Goal: Does a series of positive terms converge or diverge? ?1
kka
Ratio Test (Does ratio of successive terms approach some limit L? Then series is close to being geometric.)
,
,
,
1
1
1
lim 1
L
L
L
a
a
n
n
n
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Root Test (Does nth root of terms approach some limit L? Then series is close to being geometric.)
,
,
,
1
1
1
lim
L
L
L
ann
n
2 ln
1
nnn
Example 1 – Converge/Diverge?
13
nn n
ln
1lim 0 sequence converges to zero.
No info from Divergence Test.
1
2
1n
n
n
k
Example 2
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For what values k does the series converge?
n
n
n n
k2
1lim
1
nn
n n
k/12
1lim
n
n n
k
1lim
n
kn
n1lnlim
2
3
3nn
n
Example 3 – Converge/Diverge?
15
nn
n
3lim
3
0 sequence converges to zero.
No info from Divergence Test.
0
2
! 2
!
k k
k
Example 4 – Converge/Diverge?
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1.
2.
Direct Comparison:
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Lecture 27 – Comparison Tests
N a
1N a2N a
1N N
N a
1N a2N a1N N 1N
converges. then converges, and If
Nn
nNn
nnn abba
diverges. then diverges, and If
Nn
nNn
nnn abba
N b1N b
2N b
N b
1N b 2N b
Limit Comparison:
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terms.positive with series be and Let
Nnn
Nnn ba
diverge.both or convergeboth
and then 0,withlim Ifn
Nn
nNn
nn
n ba LL b
a
1 15
5
n n
Example 1 – Converge/Diverge?
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15
5lim
n
n0 sequence converges to zero.
No info from Divergence Test.
, as n
Example 2 – Converge/Diverge?
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1 !
1
n n
!
1lim
n
n0 sequence converges to zero.
No info from Divergence Test.
:2for general,In n
:3n :4n
Example 3 – Converge/Diverge?
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22 1
1
n n
1
1lim
2n
n0 sequence converges to zero.
No info from Divergence Test.
22
1
1
1, as
nnn
Example 4 – Converge/Diverge?
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12 1
1
n n
1
1lim
2n
n0 sequence converges to zero.
No info from Divergence Test.
nnnn
11
1
1, as
22
