Convergence: nth-Term Test, Comparing Non-negative Series, Ratio Test

Convergence: nth-Term Test, Comparing Non-negative Series,

Ratio Test

Power Series and ConvergenceWe have written statements like:

But we have not talked in depth about what values of make the identity true.

Example: Investigate whether or not makes the sentence above true? ?

:

:

ln (1+.5 )=.5− .52

2+.53

3−…+ (−1 )𝑛−1 .5

𝑛

𝑛 +…

ln (1+100 )=100− 1002

2+1003

3−…+(−1 )𝑛−1 100

𝑛

𝑛 +…

Appears to Converge.

Appears to Diverge.

We need better ways to determine the values of that make the series converge.

When is a Taylor Series a Good Approximation?

Consider the Maclaurin Series for :

Investigate the partial sums of the series and compare the results to on a graph.

Window: and



Window: and



Window: and

On , , , , , ,… converges to . The polynomial series is a good approximation of on .





Window: and



Window: and

On , , , , , ,… converges to . The polynomial series is a good approximation of on.

Proving Sine ConvergesProve the Maclaurin series converges to for all real .

Inve

stig

ate

the

erro

r fun

ctio

n: 𝑅𝑛 (𝑥 )=𝑔 (𝑛+1 ) (𝑐 ) (𝑥−0 )𝑛+1

(𝑛+1 )!

|𝑅𝑛 (𝑥 )|=|𝑔 (𝑛+1 ) (𝑐 ) (𝑥−0 )𝑛+1

(𝑛+1 )! |¿|𝑔 (𝑛+1 ) (𝑐 )𝑥𝑛+1|

(𝑛+1 )!

≤|1∙ 𝑥𝑛+1|(𝑛+1 )!

≤ |𝑥|𝑛+1

(𝑛+1 )!

Investigate the limit of the last statement

lim𝑛→∞

|𝑥|𝑛+1

(𝑛+1 ) !¿0

This means that for all .

Therefore the Maclaurin series converges for all .




Window: and



Window: and

On , , , , , ,… converges to . The series is a good approximation of on .

The Convergence Theorem for Power Series

There are three possibilities for with respect to convergence:

1. There is a positive number such that the series diverges for but converges for . The series may or may not converge at either of the endpoints and .

2. The series converges for every ().3. The series converges at and diverges elsewhere

().

Radius of Convergence

The set of all values of for which the series converges is the interval of convergence.

The n-th Term TestIf , then the infinite series diverges.

OR

If the infinite series converges, then .

The converse of this statement is NOT true.If , the infinite series does not necessarily converge.

When determining if a

series converges, always use this

test first!

ExamplesUse the n-th Term Test to investigate the convergence of the series.

1.

2.

lim𝒏→∞

𝟐𝒏+𝟏𝒏 ¿𝟐 Since the limit is not 0,

the series must diverge.

lim𝒏→∞

𝟓𝟏𝟎𝒏¿𝟎

Since the limit is 0, the n-th Term Test is

inconclusive.

Just because the n-th term goes to zero does not mean the series necessarily converges. We need more tests to

determine if a series converges.

The Finney Procedure for Determining Convergence

nth-Term TestIs ?

No The series diverges.

Yes or Maybe

?

Convergent Geometric Series

The geometric series converges if and only if . If the series converges, its sum is .

Example: Determine if converges.Where a is the first term and r

is the constant ratio.

This is a geometric series with .

Since , the series converges.

This explanation would warrant full credit on the AP Test. Statements like “I know this is a __ series with __, so it diverges/converges” are acceptable.

The Finney Procedure for Determining Convergence

nth-Term TestIs ?

No The series diverges.

Yes or Maybe

Geometric Series TestIs ?

Yes

Converges to if .

Diverges if .

No

?

ExampleInvestigate the series to see if it diverges.

∑𝑛=0

∞ (𝑛 !)2

(2𝑛 )!=1+1

2+ 16+ 120

+ 170

+ 1252

+…

Using partial sums, it appears the series converges:

Similar to a Geometric Series, we can investigate the ratio between terms.

Notice the series is NOT geometric.

1

0

1 2 0.5;1

aa

2

1

1 6 0.33;1 2

aa

3

2

1 20 0.3;1 6

aa

4

3

1 70 0.29;1 20

aa

5

4

1 252 0.281 70

aa

Similar to a converging Geometric Series, the limit of the “common” ratio appears to be less than 1. According to the next theorem. This means the series converges.

The Ratio TestSuppose the limit either exists or is infinite. Then:1. If , the series converges.

2. If , the series diverges.

3. If , the test is inconclusive.

Each successive terms are getting smaller.

Each successive terms are getting larger.

This is the “cousin” test to determining if a Convergent Geometric Series test. It is identical except: (1) If the “common” ratio is 1, then test fails. (2)

The test does not determine the value the series converges to.

Example 1Investigate the series to see if it diverges.

Similar to a Geometric Series, we can investigate the ratio between terms.

1lim n

nn

aa

2

2

1 !2 1 !

!2 !

limnn

nnn

2

2

1 ! 2 !

! 2 2 !lim n n

n nn

212 2 2 1lim nn nn

2

22 1

4 6 2lim n n

n nn

14

Similar to a converging Geometric Series, this “common” ratio is less than 1. According to the Ratio Test, this means the series converges.

Example 2In #3 on Taylor Series Challenges we worked with the series . Find the radius of convergence.

2 1

2 1

22

1 23 2 1

lim 1 23 2

n

n

n nn

xn

xn

1lim n

nn

aa

2 22

22 2

3 2 2lim

3 2 1 2

nn

nnn

n x

n x

2

2

2lim

3 1n

n xn

2

2

2lim

1 3n

xnn

22

19

x

229 1x

22 9x 2 3x

The interval of convergence is

centered at 2 with a radius

convergence of 3.

Notice: Unlike the last example, this series

depends on a value of x.

The limit depends on n. Separate the n’s.

If the series converges, the

ratio is less than 1.

Example 3Find the radius of convergence for the Maclaurin series for .

2 1

2

1 !lim

!

n

nn

xnxn

1lim n

nn

aa

The series has an infinite radius.

𝑒𝑥=1+𝑥+ 𝑥2

2 !+ 𝑥

3

3 !+…+ 𝑥

𝑛

𝑛 !+…=∑𝑛=0

∞ 𝑥𝑛𝑛!

𝑒𝑥2=1+𝑥2+ 𝑥4

2 !+ 𝑥

6

3 !+…+ 𝑥

2𝑛

𝑛! +…=∑𝑛=0

∞ 𝑥2𝑛𝑛 !

2 2

2

!lim1 !

n

nn

x nn x

2

lim1n

xn

0

We know:

Thus:

Investigate the ratio:

Example 4Find the radius of convergence for the series .

1 1 1

1

1

15lim15

n n

n

n nn

n

x

x

1lim n

nn

aa

1

11

5 1lim

5 1

nn n

nn nn

x

x

1

1

5lim5

n n

n nn

xx

lim

5n

x

Investigate the ratio:

5x

5 1x

5x

The interval of convergence is centered at 0 with a radius

convergence of 5.

If the series converges, the ratio is less than 1.

Since the ratio tests uses an absolute value, the powers on -1 do not

affect the limit.

White Board ChallengeUse the Ratio Test to determine the radius of convergence for .

1lim n

nn

aa

1 111

limnnnnn

xx

21lim n n

n nn

21lim n

nn

1

The Ratio Test is inconclusive because the limit is 1. We need to use other tests to

determine if the series converges.

The Direct Comparison TestSuppose for all .

• If converges, then converges.• If diverges, then diverges.

Example: Prove converges for all real .

For all is the Taylor Series for .

Since converges for all real , converges for all real by the Direct Comparison Test.

Both series must have only positive terms.

What about Negative Terms?Does the Direct Comparison Test fail if there are negative terms? Consider the series if :

Using partial sums, it appears the series converges:

But we don’t have a test to prove the series converges.Similar to the Ratio Test, what happens when we look at the absolute value of each term:

Using partial sums, this series ALSO appears to converges:

Since the new series is less than the original, if we can prove the new series converges, then the original series must converge.

Absolute Convergence Test

If converges, then converges.

Example: Prove converges for all .Unlike the Comparison Test, this test does not require the terms to be positive.

Notice: converges by the Ratio Test:

Since converges for all real by the Direct Comparison Test, converges for all real by the Absolute Convergence Test.

Investigate 11 !

lim 1!

n

n

n

!lim1 !n

nn

1lim1n n

01lim n

nn

aa

Definition of Absolute Convergence

If converges, then converges absolutely.

Example: The Alternating Harmonic Series converges but it does not converge absolutely:

This is the divergent Harmonic Series.

Note about Absolute Convergence

If converges, then converges absolutely.

The Ratio Test uses an absolute value: . Thus, every test that converges by the Ratio Test also converges absolutely:

converges absolutely because converges by by the ratio test

The Ratio Test is incredibly strong

Convergence: nth-Term Test, Comparing Non-negative Series, Ratio Test

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