Top Banner
9.4 Comparison of Series 9.5 Alternating Series
29

9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Dec 16, 2015

Download

Documents

Elmer Rawding
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

9.4 Comparison of Series9.5 Alternating Series

Page 2: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

The Direct Comparison TestThe Boring Book Definition: (BOOO!!!)Part 1

The series diverges if there exists another

series such that bn < an and bn diverges,

then the series diverges.

Part 2

The series converges if there exists another

series such that an <bn and bn converges,

then the series converges.

1nna

1nna

1nnb

1nna

1nna

1nnb

Page 3: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

The Analogy (Yippee!!!)

Ok... Imagine for a moment that you have a MAGIC BOX (ooh! aah!) This magic box has the ability to float in the middle of the air! (wow!)

Page 4: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Where will the magic box go?

You decide that want to push this magic box somewhere... either on the ceiling or on the floor...

Page 5: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Push it to the floor

If you want to push it to the floor, your hands need to be ON TOP of the box, and you need to push downward

Page 6: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Push it to the ceiling

If you want to push it to the ceiling, you need to put your hands underneath the box, and then push upward

Page 7: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Going nowhere...

If your hands are ON TOP of the magic box, and you push UPWARD, What happens to the box? NOTHING!

Page 8: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Going nowhere, again!

If your hands are UNDERNEATH the magic box, and you push DOWNWARD, What happens to the box? NOTHING!

Page 9: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Put it all together...

For the Direct Comparison Test, you can think of the “magic box” as the series which you want to find out if it converges or diverges.

You can think of your hands as the other series

Pushing downward means your converges

Pushing upward means your diverges

1nna

1nnb

1nnb

1nnb

Page 10: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

The Direct Comparison Test

Part 1 (rewritten)If you want to show that converges: This is like trying to push the magic box down to the floor.

You need to have two things1. You need to have your hands (bn) on top of the “magic box” (an) Mathematically, this means that bn > an

2. You need to have your hands ( ) push downward Mathematically, this means has to converge

If you can find bn that does this, then converges

1nna

1nnb

1nnb

1nna

Page 11: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

The Direct Comparison Test

Part 2 (rewritten)If you want to show that diverges, This is like trying to push the magic box up to the ceiling.

You need to have two things1. You need to have your hands (bn) underneath the

“magic box” (an)

Mathematically, this means that bn < an

2. You need to have your hands ( ) push upwardMathematically, this means has to diverge

If you can find a bn that does this, then diverges

1nna

1nna

1nnb

1nnb

Page 12: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

The Direct Comparison Test

LimitationsRemember that if you have your hands ON TOP of

the magic box and you push upward, nothing happens... Similarly:

• If you have a bn > an and diverges, then nothing happens... you don’t prove anything

If you have your hands UNDERNEATH the magic box and you push downward, nothing happens... Similarly:

• If You have a bn > an and diverges, then nothing happens... you don’t prove anything

1nnb

1nnb

Page 13: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example:Let’s try to show that converges.Let’s place in our magic box

If you want to show that converges, then we need to put our hands on top of it and push downward...

12 2

1

n nn

12 2

1

n nn

12 2

1

n nn

Page 14: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example: continued...Let’s try to choose something SIMPLE that (we

already know) converges... Let’s choose

This new series that we chose will become our “hands”. So now, we have to figure out if our “hands” go ON TOP or UNDERNEATH the MAGIC BOX.

If we plot the graphs of and

We can easily see that the graph of is on top of

12

1

n n

)( 1

)(2

handsn

nf

)( 2

1)(

2box magic

nnnf

2

1

n nn 2

12

Page 15: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example: continued...

Since the graph of our “hands” is on top of the graph of “the magic box, this means that we can place “our hands” on top of the magic box

And since we know ahead of time that converges, this means that we have everything we need to push the magic box downward... thus proving that converges!

12 2

1

n nn

12

1

n n

Page 16: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example 2

1 32

1

nn

Page 17: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example 3

1 2

1

n n

Page 18: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Limit Comparison Test

converge.both or divergeboth either

and Then positive. and finite is L where

lim

and ,00 that Suppose

nn

n

n

n

nn

ba

Lb

a

,ba

Page 19: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Examples

ban

1 )1(

Page 20: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Examples

ban

1 )1(

Page 21: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Works Great When You Have a Mess!

Compare “messy” algebraic series with p-series

35

2

2

4

10

23

1

543

1

nn

n

n

nn

Page 22: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Examples

1 )2(

2n

n

Page 23: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Examples

14

2 )3(

3n

n n

Page 24: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Alternating Series Test

naaa

aa

.a

nnnn

nn

nn

n

(2) and 0lim (1)

if converge

)1( and )1(

series galternatin The 0Let

1

1

Page 25: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example

1

1 11 )1(

n

n

n

Page 26: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example

112

)2(n

n

n

Page 27: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example

1

1 11 )3(n

n

n

n

Page 28: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Example

...4

1

4

2

3

1

3

2

2

1

2

2

1

1

1

2 )4(

Page 29: 9.4 Comparison of Series 9.5 Alternating Series. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there.

Alternating Series Remainder

1

1

is by sum theingapproximatin involved

remainder theof valueabsolute then the

condition thesatisfies series galternatin convergent a If

nnn

n

nnn

aRSS

SS

Raa

1

1

!

11

:

n

n

n

Example