1 lecture 28 – alternating series test goal: does a series (of terms that alternate between...
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Lecture 28 – Alternating Series Test
Goal: Does a series (of terms that alternate between positive and negative) converge or diverge?
13
21
11
n
n
n
n
1
11
k
k
k
1
2
11n
kn
k
e
...126
25
65
16
28
9
9
4
2
1
...5
1
4
1
3
1
2
11
...251694
5432
eeee
e
1
11n
nn a ...
8
1
3
1
4
1
2
1
2
11
2
Alternating Series Test
A series of the form with
is called alternating and converges if:
1
11n
nn a 0na
Example 1 – Converge/Diverge?
3
1
lim3
2
n
n
n
1)(
3
2
x
xxf
Thus,
Nna
aa
n
nn
allfor 0lim
.2
.1
n
1
13
21
11
n
n
n
n
is f Since
Example 2 – Converge/Diverge?
4
n
n
1lim
Nna
aa
n
nn
allfor 0lim
.2
.1
n
1
1
11
k
k
k
Example 3 – Converge/Diverge?
5
2
limn
e
n
n
Nna
aa
n
nn
allfor 0lim
.2
.1
n
1
1
2
11k
kk
k
e
Example 4 – Converge/Diverge?
6
n
na lim
Nna
aa
n
nn
allfor 0lim
.2
.1
n
1
1
11n
nn a ...
8
1
3
1
4
1
2
1
2
11
even is if
odd is if
2
11
2
2/n
n na
n
n
Using Sn (nth partial sum) to approximate S
7
n R
...)1()1(... 121
4321
nn
nn aaaaaaS
...)1( 12
n
nn aS
The remainder term when using Sn:
Example 5
8
How many terms are needed to approximate the series and yield a remainder less than 0.0001?
...36
1
25
1
16
1
9
1
4
11
12
11
n
n
n
Absolutely Converge, Conditionally Converge, or Diverge
9
.absolutely converges then converges, If nn bb
1
1
1
1n
nn
nn ab
diverges. then diverges, If nn bb
lly.conditiona convergesthen
diverges, and converges If
n
nn
b
bb
Example 6 – AC, CC, D?
10
Look at absolute value:
12
11
n
n
n
12
1
n n
2
1lim
n
n
Example 7 – AC, CC, D?
11
Look at absolute value:
1
11
n
n
n
1
1
n n
n
n
1lim
Example 8 – AC, CC, D?
12
Look at absolute value:
1
11 tan1n
n n
1
1tann
n
n
n
1tanlim
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