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9.5 Part 1 Ratio and Root Tests 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

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Page 1: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

9.5 Part 1 Ratio and Root Tests9.5 Part 1 Ratio and Root Tests Absolute ConvergenceAbsolute Convergence

Page 2: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

diverges

You’ve all seen the Harmonic Series

01

lim nn

which only proves that a seriesmight converge.

which

1

1

n nconverges

?

What if we were to alternate the signs of each term?

despite the fact that

...6

1

5

1

4

1

3

1

2

1

1

1

How would we write this using Sigma notation?

Page 3: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

This is called anThe signs of the terms alternate.

Does this series converge?.

Notice that each addition/subtraction is partially canceled by the next one.

4

1

3

1

6

1

5

1

…smaller and smaller…

The series converges as do all Alternating Series whose terms go to 0

Alternating Series

Page 4: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Alternating Series The signs of the terms alternate.

This series converges by the Alternating Series Test.

This is called the Alternating Harmonic Series which as you can see converges…unlike the Harmonic Series

Alternating Series Test

For the series:

1

11n

nn a

If nn aa 1 0lim n

naand

Then the series converges

Page 5: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Since each term of a convergent alternating series moves the partial sum a little closer to the limit:

is what kind of a series?

1

1

3

1

n

n Very good! A geometric series.

What is the sum of this series? 75.04

3

Try adding the first four terms of this series:

27

1

9

1

3

11

3

1

1

1

n

n

740741.27

20

Which is not far from 0.75…but how far?

Page 6: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Since each term of a convergent alternating series moves the partial sum a little closer to the limit:

Which means what?

27

1

9

1

3

11

3

1

1

1

n

n

Which is not far from 0.75…but how far?

108

175.740741.

...729

1

243

1

81

1

3

1

108

1

5

1

n

n

Can we agree that ?81

1...

729

1

243

1

81

1

740741.27

20

Page 7: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.

Which means that for an alternating series…

leads us to this:81

1...

729

1

243

1

81

1

San

nn

1

1

If k terms were generated then the error would be no larger than term k + 1

k

k

nn

n Sa 1

1 where k < and

then 1 kk aSS in which is the truncation error

SSk

Page 8: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

For example, let’s go back to the first four terms of our series:

What is the maximum error between this approximation and the actual sum of infinite terms?

Answer: 81

1

27

1

9

1

3

11

3

1

1

1

n

n

Which is bigger than the actual error which we determined to be 108

1

leads us to this:81

1...

729

1

243

1

81

1

Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.

Page 9: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Since each term of a convergent alternating series moves the partial sum a little closer to the limit:

That is why it is commonly referred to as the Error Bound

Answer: 81

1 So what does this answer tell us? It simply means that the error will be no larger than

81

1

Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.

Page 10: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Since each term of a convergent alternating series moves the partial sum a little closer to the limit:

This is also a good tool to remember because it is easier than the LaGrange Error Bound…which you’ll find out about soon enough…

Muhahahahahahaaa!

Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.

Page 11: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge. This is called the RATIO TEST

if the series converges.1L

if the series diverges.1L

if the series may or may not converge.1L

nnaFor the series L

a

a

n

n

n

1limif then

This test is ideal for factorial terms

Page 12: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

This also works for the nth root of the nth term. This is called the ROOT TEST(Ex. 57 pg. 508)

if the series converges.1L

if the series diverges.1L

if the series may or may not converge.1L

nnaFor the series Lan

nn

limif then

Page 13: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

If the series formed by taking the absolute value of each term converges, then the original series must also converge.

Absolute Convergence

If converges, then we say converges absolutely.na na

If converges, then converges.na na

“If a series converges absolutely, then it converges.”

Page 14: 9.5 Part 1 Ratio and Root Tests Absolute Convergence Absolute Convergence

Conditional Convergence

If diverges, then diverges.nana

If converges but diverges, then we say that

na nana converges conditionally

The Alternating Harmonic Series is a perfect example of Conditional Convergence