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Understanding the Comparison Tests in Calculus III: An Essential Guide

This resource provides a comprehensive overview of the Comparison Tests used in Calculus III, focusing on the Comparison Test and the Limit Comparison Test. It includes key concepts like the convergence and divergence of series with positive terms, comparisons with standard series, and practical examples that illustrate how to apply these tests effectively. Whether you're preparing for a bonus event or seeking to deepen your understanding of series convergence, this guide is structured to enhance your learning and performance.

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Understanding the Comparison Tests in Calculus III: An Essential Guide

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  1. MAT 1236Calculus III Section 11.4 The Comparison Tests http://myhome.spu.edu/lauw

  2. HW …. • WebAssign 11.4

  3. Bonus Event (5/9) • Your feedback is very helpful to the speaker.

  4. Bonus Event • Friday 5/23; 5:10- 5:40 (Hedging), 5:45-6:15 • You will get 2 points for the first presentation and 1 point for the second one. • If you participated in the 5/9 event, you will get an additional bonus point for attending both presentations. • You are not eligible to sign up if you signed up but did not show up last time. However, stand-by is welcome if seats are available.

  5. Reminder to Wai • Do the first 3 examples on the board without erasing • Overwrite on the first 3 examples to save time

  6. Preview • Comparison Test • Limit Comparison Test • Only work for series with positive terms • Compare with standard series • The nature of the comparison is term-by-term

  7. Examples Comp. Test Limit Comp. Test

  8. Examples Comp. Test Limit Comp. Test

  9. How does it diverge?

  10. The Comparison Test Suppose for all . If is convergent then is also convergent If is divergent then is also divergent

  11. Be Careful!!!!! Suppose for all . If is convergent then is also convergent However, If is convergent then there isno conclusion for

  12. Example 1

  13. Series to compare with • Geometric Series • Harmonic Series • -series

  14. Remarks For convenience, we will call the following series a p-series. It has the same convergence as

  15. Series to compare with • Geometric Series • Harmonic Series • -series

  16. Example 1 • Write down the general terms of the two series • Write down the comparison and range • Find the convergence of the comparison series • Make the conclusion by quoting the name of the comparison test

  17. Common Mistake/Misconception Comparing the series instead of comparing the terms STOP The series are not comparable unless you first show that they are both convergent. The comparison test is based on the comparison of general terms.

  18. Examples Comp. Test Limit Comp. Test

  19. Example 2

  20. Examples Comp. Test Limit Comp. Test

  21. Example 3

  22. Examples Comp. Test Limit Comp. Test ?

  23. The Limit Comparison Test (L.C.T.) Suppose If then both , converge or diverge

  24. Example 4

  25. Examples Comp. Test Limit Comp. Test ?

  26. Example 5

  27. Examples Comp. Test Limit Comp. Test ? ?

  28. Example 6

  29. Examples Comp. Test Limit Comp. Test ? ?

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