alternating series; conditional convergence objective: find limits of series that contain both...
TRANSCRIPT
Alternating Series; Conditional Convergence
Objective: Find limits of series that contain both positive and
negative terms.
Alternating Series
• Series whose terms alternate between positive and negative, called alternating series, are of special importance. Some examples are:
...11
)1( 41
31
21
1
k
k
k...1
1)1( 4
131
21
1
1
k
k
k
Alternating Series
• In general, an alternating series has one of the following two forms
where the aks are assumed to be positive in both cases.
...)1( 43211
1
aaaaak
kk ...)1( 4321
1
aaaaak
kk
Alternating Series
• The following theorem is the key result on convergence of alternating series.
Example 1
• Use the alternating series test to show that the following series converge.
(a) (b)
1
1
)1(
3)1(
k
k
kk
k
1
1 1)1(
k
k
k
Example 1
• Use the alternating series test to show that the following series converge.
(a) (b)
(a) This looks like the divergent harmonic series. However, this series converges since both conditions in the alternating series test are satisfied.
1
1
)1(
3)1(
k
k
kk
k
1
1 1)1(
k
k
k
1
111
kk
aa kk
01
limlim k
ak
kk
Example 1
• Use the alternating series test to show that the following series converge.
(a) (b)
(b) This series converges since both conditions in the alternating series test are satisfied.
1
1
)1(
3)1(
k
k
kk
k
1
1 1)1(
k
k
k
decreasing
1)6()4(
4
3
)1(
)2)(1(
42
21
kkk
kk
k
kk
kk
k
a
a
k
k0
)1(
3limlim
kk
ka
kk
k
Approximating Sums of Alternating Series
• The following theorem is concerned with the error that results when the sum of an alternating series is approximated by a partial sum.
Approximating Sums of Alternating Series
• This is what the theorem means.
Example 2
• Later, we will show that the sum of the alternating harmonic series is
(a) Accepting this to be true, find an upper bound on the magnitude of error that results
if ln2 is approximated by the sum of the first eight terms in the series.
...1
)1...(12ln 141
31
21
kk
Example 2
• Later, we will show that the sum of the alternating harmonic series is
(a) Accepting this to be true, find an upper bound on the magnitude of error that results if ln2 is approximated by the sum of the first eight terms in the series.
As a check we look at the exact value of the error.
...1
)1...(12ln 141
31
21
kk
12.9
1|2ln| 98 as
12.059.|2ln| 8 s
Example 2
• Later, we will show that the sum of the alternating harmonic series is
(b) Find a partial sum that approximates ln2 to one decimal-place accuracy (the nearest tenth).
...1
)1...(12ln 141
31
21
kk
Example 2
• Later, we will show that the sum of the alternating harmonic series is
(b) Find a partial sum that approximates ln2 to one decimal-place accuracy (the nearest tenth).
For one decimal-place accuracy, we must choose a value of n for which |ln2 – sn| < 0.05. Since
|ln2 – sn| < an+1 , so we need to choose n so that an+1 < .05
...1
)1...(12ln 141
31
21
kk
Example 2
• Later, we will show that the sum of the alternating harmonic series is
(b) Find a partial sum that approximates ln2 to one decimal-place accuracy (the nearest tenth).
We can use our calculator to find the value of n. Doing that, we see that a20 = .05; this tells us that partial sum s19 will provide the desired accuracy. We can also solve the equation
...1
)1...(12ln 141
31
21
kk
05.1
1
n
Absolute Convergence
• The series does not fit any of the categories studied so far- it has
mixed signs but is not alternating. We will now develop some convergence tests that can be applied to such series.
...1 432 21
21
21
21
Absolute Convergence
• First, a definition.
Example 3
• Determine whether the following series converge or diverge.
(a) (b)...1 5432 21
21
21
21
21 ...1 5
141
31
21
Example 3
• Determine whether the following series converge or diverge.
(a) (b)
(a) This series of absolute values is the convergent geometric series below, so it converges absolutely.
...1 5432 21
21
21
21
21 ...1 5
141
31
21
21
21
21
21
21
21
;1
...1 5432
ra
Example 3
• Determine whether the following series converge or diverge.
(a) (b)
(b) This series of absolute values is the divergent harmonic series below, so it diverges absolutely.
...1 5432 21
21
21
21
21 ...1 5
141
31
21
...1 151
41
31
21 k
Absolute Convergence
• It is important to distinguish between the notions of convergence and absolute convergence. For example, the series below converges by the alternating series test (alternating harmonic series), yet we demonstrated that it does not converge absolutely.
...1 51
41
31
21
Absolute Convergence
• The following theorem shows that if a series converges absolutely, then it converges.
Example 4
• Show that the following series converge.
(a) (b)
12
cos
k k
k...1 5432 2
121
21
21
21
Example 4
• Show that the following series converge.
(a) (b)
(a) We already showed that this series converges absolutely in example 3, so by the theorem, it converges.
12
cos
k k
k...1 5432 2
121
21
21
21
Example 4
• Show that the following series converge.
(a) (b)
(b) We know that the cosine function will change signs, but not in an alternating fashion, so it is not an alternating series. Testing for absolute convergence, we see that it does, so the series converges.
12
cos
k k
k...1 5432 2
121
21
21
21
22
1cos
kk
k
Conditional Convergence
• Although the last theorem is a useful tool for series that converge absolutely, it provides no information about the convergence or divergence of a series that diverges absolutely.
Conditional Convergence
• Although the last theorem is a useful tool for series that converge absolutely, it provides no information about the convergence or divergence of a series that diverges absolutely. For example the two series
both diverge absolutely since the series of absolute values is the divergent harmonic series.
...1
)1(...1 141
31
21
kk ...
1...1 4
131
21
k
Conditional Convergence
• Although the last theorem is a useful tool for series that converge absolutely, it provides no information about the convergence or divergence of a series that diverges absolutely. For example the two series
both diverge absolutely since the series of absolute values is the divergent harmonic series. However, the first series converges by the alternating series test and is said to converge conditionally; the second is a constant times the divergent harmonic series, so it diverges.
...1
)1(...1 141
31
21
kk ...
1...1 4
131
21
k
Ratio Test for Absolute Convergence
• Although one cannot generally infer convergence or divergence, the following variation of the ratio test provides a way of deducing divergence from absolute divergence in certain situations.
Ratio Test for Absolute Convergence
• Although one cannot generally infer convergence or divergence, the following variation of the ratio test provides a way of deducing divergence from absolute divergence in certain situations.
Example 5
• Use the ratio test for absolute convergence to determine whether the series converges.
(a) (b)
1 3
)!12()1(
kk
k k
1 !
2)1(
k
kk
k
Example 5
• Use the ratio test for absolute convergence to determine whether the series converges.
(a) (b)
(a)
1 3
)!12()1(
kk
k k
1 !
2)1(
k
kk
k
absolutely converges
101
2lim
2
!
)!1(
2lim
||
||lim
11
k
k
ku
ukk
k
kk
k
k
Example 5
• Use the ratio test for absolute convergence to determine whether the series converges.
(a) (b)
(b)
1 3
)!12()1(
kk
k k
1 !
2)1(
k
kk
k
diverges
kk
k
k
u
uk
k
kkk
k
k
3
)2)(12(lim
)!12(
3
3
)!12(lim
||
||lim
11
Homework
• Page 673-674
• 1-37 odd
• Please refer to page 672 for a summary of the Convergence Tests we have studied. Make sure you understand and can use all of them. Think about when you would use each test. Some are for certain situations only.