# Solutions to Problem Set 19

Math 211-03

10-24-2017

[Direct comparison]

1. Use direct comparison to determine whether the series converges or diverges.

converges, because it's 5 times a p-series with . Therefore, converges by direct comparison.

2. Use direct comparison to determine whether the series converges or diverges.

diverges, because it's 3 times a p-series with . Therefore, diverges by direct comparison.

3. Use direct comparison to determine whether the series converges or diverges.

diverges, because it's 2 times a p-series with . Therefore, diverges by direct comparison.

4. Use direct comparison to determine whether the series converges or diverges.

converges, because it's a geometric series with ratio and . Therefore, converges by direct comparison.

5. Use direct comparison to determine whether the series converges or diverges.

converges, because it's a geometric series with ratio and . Therefore, converges by direct comparison.

6. Use direct comparison to determine whether the series converges or diverges.

In particular, is positive, so the series has positive terms. The inequality above also shows that

converges, because it's 5 times a p-series with . Therefore, converges by direct comparison.

7. Use direct comparison to determine whether the series converges or diverges.

diverges, since it's a p-series with . Therefore, diverges by direct comparison.

8. Use direct comparison to determine whether the series converges or diverges.

diverges, because it's a geometric series with ratio , and . Therefore, diverges by direct comparison.

All action is involved in imperfection, like fire in smoke. - The Bhagavad Gita

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