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*

Copyright 2017 by Bruce Ikenaga