1. MAT 1236 Calculus III Section 11.6 Absolute Convergence and the Ratio and Root Tests http://myhome.spu.edu/lauw

2. HW and... WebAssign 11.6 Part II (Please take you day off to study for the final exam!)

3. Preview We want tests that work for general series Define Absolute Convergence Define Conditional Convergence Abs. Convergent implies Convergent Ratio/Root Tests (No requirement on the sign of the general terms of the series)

4. Definition

5. Example 1

6. Theorem

10. Example 1 Revisit

11. Example 1(More or Less…) Converges. Why? Converges

12. Example 1(More or Less…)

14. Example 1 The phrase used here is long, we are going to replace it by

15. T or F?

16. Definition

20. Example 2

21. Expectations Important Details: Write down the general terms Take the limit of the abs. value of the ratio of the general terms Clearly mark the criterion Make the conclusion by using the Ratio Test

22. Example 3 Note that: because

23. Example 4

24. Example 5 (Ratio/Root tests fail)

26. Example 5 No conclusion from the Ratio Test If Ratio Test fails, then Root Test will fail too

27. Example 5 Plan: Use limit comparison test to show that the series is absolutely convergent. That is, we are going to show that the series is convergent. Then is (abs.) convergent

28. PPFTNE Why not use the comparison test directly on the series?

29. Justification

30. General Situation... In the exam, you will be ask to figure out the convergence of series. There are many tests that you can use. How are you going to approach such a problem? Is there a best way to do this?

31. 18-Point Decision Chart Challenge Design a decision chart that describe the best problem solving approach. These type of charts are commonly used to visualize ideas about procedures and/or causal effects.

32. Examples

35. 18-Point Decision Chart Challenge This is to encourage you to think through the problem solving process. A maximum of three 6 points for the final exam will be awarded. Individual and teams are welcome. A winning team will share the 6 points.

36. 18-Point Decision Chart Challenge The decision chart will be judged by Accuracy and completeness Creativeness and design Must be software generated charts. Deadline: 6/1 Monday at 5pm. Must be original, do not copy from the web!