2. 9.5 Testing for Convergence

3. Remember: The series converges if. The series diverges if. The test is inconclusive if. The Ratio Test: If is a series with positive terms and then:

4. This section in the book presents several other tests or techniques to test for convergence, and discusses some specific convergent and divergent series.

5. The series converges if. The series diverges if. The test is inconclusive if. Nth Root Test: If is a series with positive terms and then: Note that the rules are the same as for the Ratio Test.

6. example: ?

7. Indeterminate, so we use L’Hôpital’s Rule formula #104 formula #103

8. example: it converges ?

9. another example: it diverges

10. Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve: This leads to: The Integral Test If is a positive sequence and where is a continuous, positive, decreasing function, then: and both converge or both diverge.

11. Example #1 Does converge? Since the integral converges, the series must converge. (but not necessarily to 2.)

12. Apply the integral test to the series: for 3 conditions: Letthen, check Example #2

13. Apply the integral test to the series: 1. f(x) positive√ 2. f(x) continuous √ 3. f(x) decreasing √ Example #2

14. Apply the integral test to the series: Integrate: Example #2

15. Apply the integral test to the series: Conclusion? Example #2

16. Apply the integral test: Hint: converges The integral converging to doesn’t mean the series converges to the same value. Try This

17. Problems with the Integral Test: You must show that f is positive, continuous and decreasing for. You must be able to find the anti-derivative.

18. p-series Test converges if, diverges if. We could show this with the integral test. If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

19. Does the following series converge or diverge? Justify your answer. converges It is a p -series with p >1 Try This

20. the harmonic series: diverges. (It is a p-series with p=1.) It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series.

21. Limit Comparison Test If and for all ( N a positive integer) If, then both and converge or both diverge. If, then converges if converges.If, then diverges if diverges.

22. Use the Limit Comparison Test to compare a series with unknown behavior with a series with known behavior. Often you can select a p- series for comparison. Important Ideas When selecting a p -series, use the same degree numerator and denominator as the given series and disregard all but the highest powers of n.

23. Example 3a: When n is large, the function behaves like: Since diverges, the series diverges. harmonic series

24. Example 3b: When n is large, the function behaves like: Since converges, the series converges. geometric series

25. Stop here for today… Assignment p523 1,2,5-15 odd

26. Alternating Series The signs of the terms alternate. Good news! example: This series converges (by the Alternating Series Test.) If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series Test This series is convergent, but not absolutely convergent. Therefore we say that it is conditionally convergent.

27. Determine the divergence or convergence of: Does ? Is ? Example:

28. n =1 n =2 n =3 n =4 n =5 n =6 n =7 n =8 Is the alternating sequence converging?

29. Determine the convergence or divergence of Diverges Try This

30. fails the alternating series test for convergence. Does this mean it diverges? How can you prove that it diverges? Yes n- th term test Try This

31. Since each term of a convergent alternating series moves the partial sum a little closer to the limit: Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. This is a good tool to remember, because it is easier than the LaGrange Error Bound.

32. Approximate the sum of the following convergent, alternating series by evaluating the first 3 partial sums. Estimate the error. Example:

33. Error < Example

34. For a power series whose radius of convergence is a finite R, there are endpoints which may be either convergent or divergent. c RR convergent ( ) divergent ENDPOINT TESTING

35. For a power series whose radius of convergence is a finite R, there are endpoints which may be either convergent or divergent. c RR convergent [ ) divergentconvergent

36. For a power series whose radius of convergence is a finite R, there are endpoints which may be either convergent or divergent. c RR convergent ( ] divergentconvergent

37. For a power series whose radius of convergence is a finite R, there are endpoints which may be either convergent or divergent. c RR convergent [ ]

38. Find the interval of convergence Hint: find the radius of convergence using the Ratio Test then test the endpoints

39. Find the interval of convergence. Be sure to test endpoints. [-1,1] Try This

40. Assignment : p523:17-20 all, 35a, 37a, 41,a