2. Infinite Series 9 Copyright © Cengage Learning. All rights reserved.

3. The Ratio and Root Tests Copyright © Cengage Learning. All rights reserved. 9.6

4. Use the Ratio Test to determine whether a series converges or diverges. Use the Root Test to determine whether a series converges or diverges. Review the tests for convergence and divergence of an infinite series. Objectives

5. The Ratio Test

7. Why does the ratio test work?

8. The series converges by the ratio test.

9. This series diverges by the ratio test.

10. The series converges by the ratio test.

11. Ex: Do the terms go to zero? This series converges absolutely by the Ratio Test.

12. Ex: #36 This series converges absolutely by the Ratio Test. Try this one on your own

13. The Root Test

15. Why does this work?

16. This series diverges by the root test.

17. This series converges by the root test.

18. Ex: Do the terms go to zero? Beats me… Let’s go straight to the Root Test. This series converges absolutely by the Root Test.

19. Strategies for Testing Series

21. Example 5 – Applying the Strategies for Testing Series Determine the convergence or divergence of each series.

22. a. For this series, the limit of the nth term is not 0. So, by the nth-Term Test, the series diverges. b.This series is geometric. Moreover, because the ratio (r = π/6) of the terms is less than 1 in absolute value, you can conclude that the series converges. c.Because the function is easily integrated, you can use the Integral Test to conclude that the series converges. d. The nth term of this series can be compared to the nth term of the harmonic series. After using the Limit Comparison Test, you can conclude that the series diverges. Example 5 – Solution

23. e.This is an alternating series whose nth term approaches 0. Because you can use the Alternating Series Test to conclude that the series converges. f.The nth term of this series involves a factorial, which indicates that the Ratio Test may work well. After applying the Ratio Test, you can conclude that the series diverges. g. The nth term of this series involves a variable that is raised to the nth power, which indicates that the Root Test may work well. After applying the Root Test, you can conclude that the series converges. cont’d Example 5 – Solution

24. Handout: Strategies for Testing Series

25. cont’d Strategies for Testing Series

26. cont’d Strategies for Testing Series

27. Series Susan Cantey © 2007 Series are serious business; the terms are the first thing you check, If they don’t go to zero, divergence is a sure bet, Alternating series are the next best, ‘cause if the terms (decreasing) go to zero, they will converge, Moreover the error in S sub n is less than the very next term. Positive series are tougher; you’ll need a bevy of tests, The terms need to shrink so much faster sometimes it’s hard to know which test is best, P-series, p-series, one over n to the power of p, They will converge if p’s greater than one; if not it’s a divergent sum. If last comes to last, you can always compare, Your series to one from the past, If the limit of their ratio has a positive value, What the old one must do, the new one will too! Factorials necessitate the ratio test; exponents require the root test, One is the cut off for both of them; convergence when the limit is less, Or you can make a sub n into a function and its integral examine, If you get a real number positive result then the series convergences to some unknown sum.

28. HOMEWORK: Section 9.6, pg.645: 13-33 odd, 37-67 odd