MAT 1236Calculus III

Section 11.6

Absolute Convergence and the Ratio and Root Tests

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HW and ...

WebAssign 11.6 Part II (Please take you day off to study for the

final exam!)

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We want tests that work for general series Define Absolute Convergence Define Conditional Convergence Abs. Convergent implies Convergent Ratio/Root Tests (No requirement on the

sign of the general terms of the series)

Definition

is absolutely convergent if

is convergent

The point: Absolute convergence may be easier to show, because …

Example 1

1

21 1

)1(n

n

n

12

1

1( 1)n

n n

Theorem

If is absolutely convergent

then is convergent

Theorem

If is absolutely convergent

then is convergent

OR equivalently

If is convergent

then is convergent

Theorem

If is absolutely convergent

then is convergent

OR equivalently

If is convergent

then is convergent

The point: To show a series is convergent, it suffices to show that it is abs. convergent.

Theorem

If is absolutely convergent

then is convergent

OR equivalently

If is convergent

then is convergent

Example 1 Revisit

12

1

1( 1) is absolutely convergentn

n n

Example 1(More or Less…)

Converges.

Why?

2 2

12 2 2 2 2

1

12 2 2 2 2

1

1 1 2 1 0 0

3 5

1 1 1 1 1( 1) 1

2 3 4 5

1 1 1 1 1 ( 1) 1

2 3 4 5

n

n

n

n

C

An

Bn

B C A

Converges

Example 1(More or Less…)

2 2

12 2 2 2 2

1

12 2 2 2 2

1

1 1 2 1 0 0

3 5

1 1 1 1 1( 1) 1

2 3 4 5

1 1 1 1 1 ( 1) 1

2 3 4 5

n

n

n

n

C

An

Bn

B C A

Example 1(More or Less…)

2 2

12 2 2 2 2

1

12 2 2 2 2

1

1 1 2 1 0 0

3 5

1 1 1 1 1( 1) 1

2 3 4 5

1 1 1 1 1 ( 1) 1

2 3 4 5

n

n

n

n

C

An

Bn

B C A

convergent thereforeand convergent absolutely is 1

)1(

)12 series,-( convergent is 11

)1(

12

1

12

12

1

n

n

nn

n

n

ppnn

Example 1

The phrase used here is long, we are going to replace it by

1

21 1

)1(n

n

n

convergent (abs.) is 1

)1(1

21

n

n

n

T or F?

If is not absolutely convergent

then is divergent.

Definition

is conditionally convergent if

is convergent but not abs. convergent

Definition

is conditionally convergent if

is convergent but not abs. convergent

Series

ries Se

Convergent Abs.

eries S

Convergent Cond.

Ratio Tests for

Divergent,1

Conclusion No1

Convergent (Abs.)1

limTestRoot

limTest Ratio

1

L

L

L

a

a

a

nn

n

n

n

n

Ratio/Root Tests for

Divergent,1

Conclusion No1

Convergent (Abs.)1

limTestRoot

limTest Ratio

1

L

L

L

a

a

a

nn

n

n

n

n

Example 2

1

2

2)1(

nn

n n

11 (Abs.) Convergent

lim 1 No Conclusion

1, Divergent

n

nn

La

La

L

Expectations

Important Details: Write down the general terms Take the limit of the abs. value of the

ratio of the general terms Clearly mark the criterion Make the conclusion by using the Ratio

Test

Example 3

Note that:

because

1 )!3(

1

n n

123)2)(1)((3)!(3

123)23)(13)(3()!3(

)!(3)!3(

nnnn

nnnn

nn

11 (Abs.) Convergent

lim 1 No Conclusion

1, Divergent

n

nn

La

La

L

Example 4

1 32

1

n

n

n

n

1 (Abs.) Convergentlim

1 No Conclusion

1, Divergent

nn

n

La

L

L

Example 5 (Ratio/Root tests fail)

14

1

2)1(

n

nn

n

Example 5 (Ratio/Root tests fail)

14

1

2)1(

n

nn

n

1 11 1

1 44

11 41

14 1

4

41 1 1 1

1 1

( 1) 2 ( 1) 2;

1

( 1) 2lim lim

1 ( 1) 2

1lim 2 lim 2

11 1

1

n nn n

n n

n nn

n nnn n

n n n n

n n

a an n

a n

a n

n

nn

Example 5

14

1

2)1(

n

nn

n

No conclusion from the Ratio Test If Ratio Test fails, then Root Test will fail

too

Example 5

14

1

2)1(

n

nn

n

Plan: Use limit comparison test to show that the series is absolutely convergent.

That is, we are going to show that the series

is convergent.

Then is (abs.) convergent

14

1

14

1

22)1(

n

n

n

nn

nn

14

1

2)1(

n

nn

n

PPFTNE

Why not use the comparison test directly on the series?

14

1

2)1(

n

nn

n

Justification

You do not need to justify the following

and for

General Situation...

In the exam, you will be ask to figure out the convergence of series.

There are many tests that you can use. How are you going to approach such a problem?

Is there a best way to do this?

18-Point Decision Chart Challenge

Design a decision chart that describe the best problem solving approach.

These type of charts are commonly used to visualize ideas about procedures and/or causal effects.

Examples

Examples

Examples

18-Point Decision Chart Challenge

This is to encourage you to think through the problem solving process.

A maximum of three 6 points for the final exam will be awarded.

Individual and teams are welcome. A winning team will share the 6 points.

18-Point Decision Chart Challenge

The decision chart will be judged by• Accuracy and completeness• Creativeness and design

Must be software generated charts. Deadline: 6/1 Monday at 5pm. Must be original, do not copy from the

web!