W10 - HW Solutions

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 .

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 .

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:

04

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

05

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

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 .

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 .

08

If for , then we know for .

09

(a)

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(b)