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Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.
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Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Jan 03, 2016

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Page 1: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Alternating Series; Conditional Convergence

Objective: Find limits of series that contain both positive and

negative terms.

Page 2: 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

Page 3: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 4: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Alternating Series

• The following theorem is the key result on convergence of alternating series.

Page 5: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 6: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 7: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 8: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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.

Page 9: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Approximating Sums of Alternating Series

• This is what the theorem means.

Page 10: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 11: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 12: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 13: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 14: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 15: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 16: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Absolute Convergence

• First, a definition.

Page 17: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Example 3

• Determine whether the following series converge or diverge.

(a) (b)...1 5432 21

21

21

21

21 ...1 5

141

31

21

Page 18: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 19: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 20: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 21: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Absolute Convergence

• The following theorem shows that if a series converges absolutely, then it converges.

Page 22: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

Example 4

• Show that the following series converge.

(a) (b)

12

cos

k k

k...1 5432 2

121

21

21

21

Page 23: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 24: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 25: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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.

Page 26: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 27: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 28: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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.

Page 29: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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.

Page 30: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 31: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 32: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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

Page 33: Alternating Series; Conditional Convergence Objective: Find limits of series that contain both positive and negative terms.

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.