W10 - Homework

means: “eligible for Friday Presentation”

Stepwise problems - Thu. 11:59pm

Power series: Radius and Interval

01

01

Power series - radius and interval

Find the radius and interval of convergence for these power series:

(a) (b) (c)

Solution

01

(a)

Therefore, the radius of convergence is and the preliminary interval is .

Check end points:

Both of these series converge, so the final interval of convergence is .

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(b)

Therefore, and the preliminary interval is .

Check end points:

The first series converges by the AST. The second diverges ().

So the final interval of convergence is .

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(c)

Therefore, the radius of convergence is and the preliminary interval is .

Check end points:

Both series diverge. So the final interval is .

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Power series as functions

02

01

Modifying geometric power series

Consider the geometric power series for .

For this problem, you should modify the series for .

(a) Write as a power series and determine its interval of convergence.

(b) Write as a power series and determine its interval of convergence.

Solution

02

(a)

The geometric series for converges when and diverges for . So ours will converge when , which is when , and diverge otherwise. The interval is therefore .

One can check this in more detail by doing the ratio test:

But we must be careful: the ratio test will not tell us what happens at the endpoints of the interval. If we apply the ratio test here, we would have to check the endpoint separately. But if we use the known result for geometric series, we know it diverges at both endpoints.

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(b)

The geometric series for converges when . So our series will converge when , which is when , and diverges for . So the interval is .

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03

03

Finding a power series

Find a power series representation for these functions:

(a) (b)

Solution

03

(a)

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Another approach:

We know that:

Plug in :

Complete:

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(b)

Notice:

Integrate:

Plug in to solve and find .

Now then:

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Regular problems - Sun. 11:59pm

Series tests: strategy tips

04

01

Various limits, Part I

Find the limits. You may use or or as appropriate. Braces indicate sequences.

• C = Convergent
• AC = Absolutely Convergent
• CC = Conditionally Convergent
• D = Divergent

		C or D		C or D	AC, CC, or D	AC, CC, or D

Solution

04

		C or D		C or D	AC, CC, or D	AC, CC, or D
	0		0
	1
	0		0
	0		0
	0		0

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05

02

Various limits, Part II

Find the limits. You may use or or as appropriate. Braces indicate sequences.

• C = Convergent
• AC = Absolutely Convergent
• CC = Conditionally Convergent
• D = Divergent

		C or D		C or D	AC, CC, or D	AC, CC, or D

Solution

05

		C or D		C or D	AC, CC, or D	AC, CC, or D
	0		0
	0		0
	0		0

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Power series: Radius and Interval

06

02

Power series - radius and interval

Find the radius and interval of convergence for these power series:

(a) (b) (c)

Solution

06

(a)

Therefore and .

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(b)

Therefore and the preliminary interval is .

At we have . This diverges ().

At we have . This converges by the AST.

Therefore, the final interval of convergence is .

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(c)

Observe that so . Assume . Then:

If , then of course the series is and converges to .

Therefore and .

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07

03

Power series - radius and interval

Find the radius and interval of convergence for these power series:

(a) (b)

Solution

07

(a)

Apply the ratio test:

Therefore and the preliminary interval is .

Check endpoints:

At , we have , which converges absolutely.

At , we have , which converges by the DCT, comparing with .

Therefore, the final interval of convergence is .

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(b)

Apply the ratio test:

Therefore, and the interval of convergence is .

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Power series as functions

08

02

Power series of a derivative

Suppose that a function has power series given by:

The radius of convergence of this series is .

What is the power series of and what is its interval of convergence?

Solution

08

If for , then we know for .

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09

04

Modifying and integrating a power series

(a) Modify the power series for to obtain the power series for .

(b) Now integrate this series to find the power series for .

Solution

09

(a)

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(b)

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