9.4 Comparison of Series9.5 Alternating Series

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

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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!)

Where will the magic box go?

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

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

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

Going nowhere...

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

Going nowhere, again!

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

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

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

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

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

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1nnb

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

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12 2

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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

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2box magic

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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!

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1

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Example 2

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Example 3

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Limit Comparison Test

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lim

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Examples

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Examples

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Works Great When You Have a Mess!

Compare “messy” algebraic series with p-series

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Examples

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Alternating Series Test

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Alternating Series Remainder

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