M38 Lec 121013

7/18/2019 M38 Lec 121013

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MATH 38

Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

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Outline

1 Summary of Convergence Tests for Infinite Series

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These may come in handy...

Remark

To find the common ratio and first term of a geometric series, put it in theform ∞

n =0ar

n

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These may come in handy...

Remark

To find the common ratio and first term of a geometric series, put it in theform ∞

n =0ar

n

Example

1

∞n =0

2n +1

3n

2 ∞n =1

12n

3

∞n =0

(−1)n +15n −13n 2n +1

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Before we proceed...

True or False

Suppose limn →∞a n = 0.

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True or False


1 The sequence {a n } is convergent.

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True or False



2

The series a

n convergent.

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True or False



2

The series a

n convergent.

Remark

Know the difference between a convergent sequence and a convergent

series.

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Summary of Convergence Tests for Infinite Series

Section 1.5 of Lecture Notes

List of all convergence tests

Set A: Test for Divergence

Set B: Tests for Series of Nonnegative TermsSet C: Test for Alternating SeriesSet D: Tests for Absolute Convergence

List of all special series

Some additional tips

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Please add:

1 If a series is telescoping,

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Please add:

1 If a series is telescoping, find SPS to determine if convergent.

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Please add:

1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form

(a n ±b n ) where

a n and

b n are special

series,

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Please add:


(a n ±b n ) where

a n and

b n are special

series, use four convergence theorems:

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Please add:


(a n ±b n ) where

a n and

b n are special


If both are convergent, then (a n ±b n )

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Please add:


(a n ±b n ) where

a n and

b n are special


If both are convergent, then (a n ±b n ) is convergent.

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Please add:


(a n ±b n ) where

a n and

b n are special


If both are convergent, then (a n ±b n ) is convergent. (Know how to

get the sum!)

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Please add:


(a n ±b n ) where

a n and

b n are special


If both are convergent, then

(a n ±b n ) is convergent. (Know how to

get the sum!)If one is divergent and the other is convergent, then

(a n ±b n )

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Please add:


(a n ±b n ) where

a n and

b n are special





(a n ±b n ) is

divergent.

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Please add:


(a n ±b n ) where

a n and

b n are special





(a n ±b n ) is

divergent.

3 If you are asked to get the sum, the series must be telescoping orgeometric (for now).

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Convergent or Divergent?1 Ask first: Is the divergence test applicable? Can it be easily applied?

( limn →∞a n is easy to compute and it seems that the limit is not zero)

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2 Guess (intelligently)

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3 Determine the appropriate test to use. Set D Tests may be usedbecause if the IS is absolutely convergent, then the IS is convergent.

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3 Determine the appropriate test to use. Set D Tests may be usedbecause if the IS is absolutely convergent, then the IS is convergent.

4 NO CONCLUSION is NOT an acceptable final answer.

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Example

Determine if the following infinite series are convergent or divergent.

1

∞n =1

n +1

2n 2−1

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Example


1

∞n =1

n +1

2n 2−1

2 ∞n =1

2e

n

e n +4n

n

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Example


1

∞n =1

n +1

2n 2−1

2 ∞n =1

2e

n

e n +4n

n

3

∞n =1

e −5n + 1

n n

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Example


1

∞n =1

n +1

2n 2−1

2 ∞n =1

2e

n

e n +4n

n

3

∞n =1

e −5n + 1

n n

4

∞n =1

3

5n −3− 3

5n +2

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Example


1

∞n =1

n +1

2n 2−1

2 ∞n =1

2e

n

e n +4n

n

3

∞n =1

e −5n + 1

n n

4

∞n =1

3

5n −3− 3

5n +2

5 ∞n =2

n lnn

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Example


1

∞n =1

n +1

2n 2−1

2 ∞n =1

2e

n

e n +4n

n

3

∞n =1

e −5n + 1

n n

4

∞n =1

3

5n −3− 3

5n +2

5 ∞n =2

n lnn

6

∞n =0

1

4n +4n

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Remark

AST may not always be the best way to show that an alternating series isconvergent.

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Remark

AST may not always be the best way to show that an alternating series isconvergent. Sometimes, it is easier to show that a series with negativeterms is convergent via ABSOLUTE CONVERGENCE.

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Remark


Example

Determine whether the following infinite series are convergent or divergent.

1

∞n =0

(−1)n (2n +1)

(n +1)!

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Remark


Example

Determine whether the following infinite series are convergent or divergent.

1

∞n =0

(−1)n (2n +1)

(n +1)! 2

∞n =1

(−1)n +1 sinπ

n

n 2

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Absolutely Convergent, Conditionally Convergent or Divergent?1 Is

a n absolutely convergent?

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Is |a n | convergent? Applicable tests: A, B or D.

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2 If it is not absolutely convergent and you have not been able toconclude that it is divergent (via ratio or root test): Is it conditionallyconvergent?.

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Is

a n convergent?

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Is

a n convergent?

3 If it is not conditionally convergent, it must be divergent. Justify.

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Example

Determine whether the following infinite series are absolutely convergent,conditionally convergent or divergent.

1

∞n =1

cosn

n 2

2∞n =1

(−1)n n 10n 2+1

3

∞n =1

(−1)n n 10n +1

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Example

Determine whether∞n =1

cosn

n 2 is absolutely convergent, conditionally

convergent or divergent.

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Example


cosn



Is it absolutely convergent?

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Example


cosn




Is∞n =1

cosn n 2

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Example


cosn




Is∞n =1

cosn n 2

= ∞n =1

|cosn |n 2

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Example


cosn




Is∞n =1

cosn n 2

= ∞n =1

|cosn |n 2

convergent?

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Example


cosn




Is∞n =1

cosn n 2

= ∞n =1

|cosn |n 2

convergent? YES

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Example


cosn




Is∞n =1

cosn n 2

= ∞n =1

|cosn |n 2

convergent? YES

∴ The series is absolutely convergent.

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Example

Determine whether ∞n =1

(−1)n n

10n 2+1 is absolutely convergent, conditionally


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Example

Determine whether ∞n =1

(−1)n n




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Example

Determine whether

∞n =1

(−1)n n




Is ∞n =1

(−1)n

n 10n 2+1

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Example

Determine whether

∞n =1

(−1)n n




Is ∞n =1

(−1)n

n 10n 2+1

= ∞n =1

n 10n 2+1

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n

n 10n 2+1

= ∞n =1

n 10n 2+1

convergent?

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n

n 10n 2+1

= ∞n =1

n 10n 2+1

convergent? NO

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n

n 10n 2+1

= ∞n =1

n 10n 2+1

convergent? NO

Is it conditionally convergent?

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n n

10n 2+1

= ∞n =1

n

10n 2+1convergent? NO


Is∞n

=1

(−1)n n 10n 2

+1

convergent?

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n n

10n 2+1

= ∞n =1

n



Is∞n

=1

(−1)n n 10n 2

+1

convergent? YES

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n n

10n 2+1

= ∞n =1

n



Is∞n

=1

(−1)n n 10n 2

+1

convergent? YES

∴ The series is conditionally convergent.

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Example

Determine whether

∞n =1

(

−1)n n

10n +1 is absolutely convergent, conditionally


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Example

Determine whether

∞n =1

(

−1)n n




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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n n

10n +1

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n

10n +1

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Example

Determine whether

∞n =1

(

−1)n n




Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n

10n +1convergent?

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Example

Determine whether

∞n =1

(

−1)n n

10n +1 is absolutely convergent, conditionallyconvergent or divergent.


Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n

10n +1convergent? NO

l

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Example

Determine whether

∞n =1

(

−1)n n



Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n



E l

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Example

Determine whether

∞n =1

(

−1)n n



Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n



Is∞n

=1

(−1)n n 10n

+1

convergent?

E l

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Example

Determine whether

∞n =1

(

−1)n n



Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n



Is∞n

=1

(−1)n n 10n

+1

convergent? NO

E l

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Example

Determine whether

∞n =1

(

−1)n n



Is ∞n =1

(−1)n n

10n +1

= ∞n =1

n



Is∞n

=1

(−1)n n 10n

+1

convergent? NO

∴ The series is divergent.

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Announcement:

Chapter 1 Quiz will be on December 12, Thursday (Lecture Class)

You will be allowed to take home your SW bluebook tomorrow.

Your Chapter 1 SW Bluebook will be collected on Dec 12, duringlecture class.

M38 Lec 121013

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