Post on 08-Dec-2020
Quiz
a. Give a ‘special’ infinite series to which the following series can be compared to.
b. Tell whether the series you gave is convergent or not.
2
1
2 1
1n
n
n1.
2
11
n
n
n
e
e2.
2
1
2
3
n
n
nn
3.
1.5
ALTERNATING SERIES
An infinite series of the form
1
1n
n
n
u
Recall:
1
1
1n
n
n
u
or
where 0 ,nu n N
1 2 3 ...u u u
1 2 3 ...u u u
is called an alternating series.
Examples.
1
71n
nn
2 1
1
1n n
n
e
1
1
7
8
n
n
Which of the ff. are alternating series?
2 2
1
11 sec
n
nn
1
1
1
2 1 2 1
n
nn n
1
1n
nn
n
1
31
5 4
n
n
n
n
Theorem. Alternating Series Test
The alternating series
1
1n
n
n
u
1
1
1n
n
n
u
or
converges if both conditions are satisfied:
i. for all 1n nu u n N
ii. lim 0nn
u
Examples. Use the alternating series test
to show that the series is convergent
1.
1
2
11
ln
n
nn
i. Let ln 1 lnn n n N
ii. 1
lim 0lnn n
Note: 1
lnnu
n
1 1
ln ln 1n n
Then
Thus, is convergent by AST. 2
1
lnn
n
Examples. Use the alternating series test
to show that the series is convergent
2.
i. Let
Note:
sechnu n
21
x
x
ef x
e
1
1 sechn
n
n
2
2
1
n
n n
eu
e
Then
2
22
1'
1
x x
x
e ef x
e
0
Examples. Use the alternating series test
to show that the series is convergent
2.
Note:
sechnu n
1
1 sechn
n
n
2
2
1
n
n n
eu
e
ii. 2
2lim 0
1
n
nn
e
e
Thus, is convergent by AST.
1
1 sechn
n
n
RECALL
Test for Convergence
Test for Divergence
Direct Comparison Test (DCT)
Limit Comparison Test (LCT)
Integral Test
Alternating Series Test (AST)
nth term test
Direct Comparison Test (DCT)
Limit Comparison Test (LCT)
Integral Test
Definition.
The infinite series is said to be
1
n
n
a
a. absolutely convergent if is
convergent. 1
n
n
a
b. conditionally convergent if
is convergent but is divergent.
1
n
n
a
1
n
n
a
Examples:
1
11n
nn
is absolutely
convergent.
1
21
11n
nn
is conditionally
convergent.
Theorem.
An infinite series that is absolutely
convergent are convergent.
Examples:
3
11
1
5
n
nn
are both convergent
since both are
absolutely
convergent.
1
31
12
n
n
n
n
Other tests for convergence:
1. Absolute Ratio Test
2. Root Test
Absolute Ratio Test
Let be a series for which
and let .
1
n
n
u
1
limn
n n
uL
u
0nu
If , then the given series is
absolutely convergent. 1L
If or if , then the given
series is divergent. 1L L
If , test fails. 1L
Root Test
Let be a series for which
and let .
1
n
n
u
lim n
nn
L u
0nu
If , then the given series is
absolutely convergent. 1L
If or if , then the given
series is divergent. 1L L
If , test fails. 1L
Examples. Use either the ratio or root test
to determine if the series is convergent.
1.
20
110 n
n
n
Use: Ratio Test
1lim
n
n n
uL
u
lim
n
20
1 20
1 10
10
n
n
n
n
201 1
lim10n
n
n
1
10 1
Thus, is absolutely convergent.
20
110n
n
n
Examples. Use either the ratio or root test
to determine if the series is convergent.
2.
1
3
!
n
nn
Use: Ratio Test
1lim
n
n n
uL
u
lim
n
13 !
1 ! 3
n
n
n
n
3lim
1n n
0 1
Thus, is absolutely convergent.
1
3
!
n
nn
Examples. Use either the ratio or root test
to determine if the series is convergent.
3.
1
3
2
n
nn
Use: Root Test
lim nn
nL u
lim
n
3
2
n
n
3lim
2n
3
2 1
Thus, is divergent.
1
3
2
n
nn
END
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