# Solutions to Problem Set 21

Math 211-03

10-30-2017

[Ratio and root test]

1. Use the Ratio Test to determine whether the series converges or diverges.

In computing the limit, I used the facts that

The reason is that all four are geometric sequences with ratios between -1 and 1.

The series converges by the Ratio Test.

2. Use the Ratio Test to determine whether the series converges or diverges.

The series diverges by the Ratio Test.

3. (a) Show that the Ratio Test fails when it is applied to the series .

(b) Use Limit Comparison to determine whether the series converges or diverges.

(a)

Since the limit is 1, the Ratio Test fails.

Note: The point here is that if the general term of the series is "just powers", you should not use the Ratio Test.

(b)

The limit is a finite positive number. converges, because it's a p-series with . Therefore, the original series diverges by Limit Comparison.

4. Use the Root Test to determine whether the series converges or diverges.

The series converges by the Root Test.

5. Use the Root Test to determine whether the series converges or diverges.

Let . Then

So

Therefore,

The series converges by the Root Test.

6. Use the Root Test to determine whether the series converges or diverges.

The series converges by the Root Test.

The trouble with the rat race is that even if you win you're still a rat. - Lily Tomlin

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