MAT 1236Calculus III

Section 11.6Absolute Convergence and

the Ratio and Root Tests

http://myhome.spu.edu/lauw

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

final exam!)

Preview 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

TheoremIf is absolutely convergent then is convergent

TheoremIf is absolutely convergent then is convergentOR equivalentlyIf is convergentthen is convergent

TheoremIf is absolutely convergent then is convergentOR equivalentlyIf is convergentthen is convergentThe point: To show a series is convergent,

it suffices to show that it is abs. convergent.

TheoremIf is absolutely convergent then is convergentOR equivalentlyIf is convergentthen 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 03 5

1 1 1 1 1( 1) 12 3 4 5

1 1 1 1 1 ( 1) 12 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 03 5

1 1 1 1 1( 1) 12 3 4 5

1 1 1 1 1 ( 1) 12 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 03 5

1 1 1 1 1( 1) 12 3 4 5

1 1 1 1 1 ( 1) 12 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 convergentthen 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 SeConvergent Abs.

eries SConvergent Cond.

Ratio Tests for

Divergent,1Conclusion No1

Convergent (Abs.)1

limTestRoot

limTest Ratio

1

LLL

aa

a

nnn

n

n

n

Ratio/Root Tests for

Divergent,1Conclusion No1

Convergent (Abs.)1

limTestRoot

limTest Ratio

1

LLL

aa

a

nnn

n

n

n

Example 2

1

2

2)1(

nn

n n

1 1 (Abs.) Convergentlim

1 No Conclusion1, Divergent

n

nn

LaLa

L

ExpectationsImportant 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)!(3123)23)(13)(3()!3(

)!(3)!3(

nnnnnnnn

nn

1 1 (Abs.) Convergentlim

1 No Conclusion1, Divergent

n

nn

LaLa

L

Example 4

1 321

n

n

nn

1 (Abs.) Convergentlim

1 No Conclusion1, Divergent

nnn

La

LL

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 11 1

( 1) 2 ( 1) 2;1

( 1) 2lim lim1 ( 1) 2

1lim 2 lim 2 11 1

1

n nn n

n n

n nn

n nnn n

n n n nn n

a an n

a na n

nn

n

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

PPFTNEWhy 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!