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MAT 1236 Calculus III

MAT 1236 Calculus III. Section 11.6 Absolute Convergence and the Ratio and Root Tests. http://myhome.spu.edu/lauw. HW 11.4 #13 solutions. See Method I Bonus points for an alternative solution with “significant” difference. No Class Tomorrow. Take the time to review for the final. HW.

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MAT 1236 Calculus III

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

  2. HW 11.4 #13 solutions • See Method I • Bonus points for an alternative solution with “significant” difference.

  3. No Class Tomorrow • Take the time to review for the final

  4. HW • WebAssign 11.6

  5. Preview • We want tests that work for general series • DefineAbsolute Convergence • DefineConditional Convergence • Abs. ConvergentimpliesConvergent • Ratio/Root Tests(No requirement on the sign of the general terms of the series)

  6. Definition is absolutely convergent if is convergent The point: Absolute convergence may be easier to show, because …

  7. Example 1

  8. Theorem If is absolutely convergent then is convergent

  9. Theorem If is absolutely convergent thenis convergent OR equivalently If is convergent then is convergent

  10. Theorem If is absolutely convergent then is convergent OR equivalently If is convergent then is convergent The point: To show a series is convergent, it suffices to show that it is abs. convergent.

  11. Theorem If is absolutely convergent then is convergent OR equivalently If is convergent then is convergent WHY?

  12. Example 1 Revisit

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

  14. Example 1(More or Less…)

  15. Example 1(More or Less…)

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

  17. T or F? If is not absolutely convergent then is divergent.

  18. Definition is conditionally convergent if is convergent but not abs. convergent

  19. Definition is conditionally convergent if is convergent but not abs. convergent Convergent Series

  20. Ratio Tests for

  21. Ratio/Root Tests for

  22. Example 2

  23. 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

  24. Example 3 Note that: because

  25. Example 4

  26. Example 5 (Ratio/Root tests fail)

  27. Example 5 (Ratio/Root tests fail)

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

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

  30. Question Why not use the comparison test directly on the series?

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