# Solutions to Problem Set 20

Math 211-03

10-26-2017

[Limit comparison]

1. Use Limit Comparison to determine whether the series converges or diverges.

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

2. Use Limit Comparison to determine whether the series converges or diverges.

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

3. Use Limit Comparison to determine whether the series converges or diverges.

The limit is a finite positive number. converges, because it's a geometric series with ratio , and . Therefore, the original series converges by Limit Comparison.

4. Determine whether the series converges or diverges.

The series diverges by the Zero Limit Test.

5. Use Limit Comparison to determine whether the series converges or diverges.

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

[Ratio and root test]

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

The series converges by the Ratio Test.

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

(The limit is zero because the highest power on the top is , while the highest power on the bottom is .) The series converges by the Ratio Test.

No person is a friend who demands your silence, or denies your right to grow. - Alice Walker

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