Convergence: nth-Term Test, Comparing Non-negative Series, Ratio Test

1. Convergence: nth-Term Test, Comparing Non-negative Series, Ratio Test

2. Power Series and Convergence We have written statements like: But we have not talked in depth about what values of make the identity true. Example: Investigate whether or not makes the sentence above true? ? Appears to Converge. : Appears to Diverge. : We need better ways to determine the values of that make the series converge.

3. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

4. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

5. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

6. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

7. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

8. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

9. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : On , , , , , ,… converges to . The polynomial series is a good approximation of on . Window: and

10. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

11. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

12. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

13. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

14. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

15. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

16. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : On , , , , , ,… converges to . The polynomial series is a good approximation of on. Window: and

17. Proving Sine Converges Prove the Maclaurin series converges to for all real . Investigate the limit of the last statement Investigate the error function: This means that for all . Therefore the Maclaurin series converges for all .

18. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

19. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

20. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

21. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

22. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

23. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : Investigate the partial sums of the series and compare the results to on a graph. Window: and

24. When is a Taylor Series a Good Approximation? Consider the Maclaurin Series for : On , , , , , ,… converges to . The series is a good approximation of on . Window: and

25. The Convergence Theorem for Power Series There are three possibilities for with respect to convergence: • There is a positive number such that the series diverges for but converges for .The series may or may not converge at either of the endpoints and . • The series converges for every (). • The series converges at and diverges elsewhere (). Radius of Convergence The set of all values of for which the series converges is the interval of convergence.

26. The n-th Term Test If , then the infinite seriesdiverges. OR If the infinite series converges, then . When determining if a series converges, always use this test first! The converse of this statement is NOT true. If , the infinite series does not necessarily converge.

27. Examples Use the n-th Term Test to investigate the convergence of the series. Since the limit is not 0, the series must diverge. Since the limit is 0, the n-th Term Test is inconclusive. Just because the n-th term goes to zero does not mean the series necessarily converges. We need more tests to determine if a series converges.

28. The Finney Procedure for Determining Convergence nth-Term Test Is ? No The series diverges. Yes or Maybe ?

29. Convergent Geometric Series The geometric series converges if and only if . If the series converges, its sum is . Example: Determine if converges. Where a is the first term and r is the constant ratio. This is a geometric series with . Since , the series converges. This explanation would warrant full credit on the AP Test. Statements like “I know this is a __ series with __, so it diverges/converges” are acceptable.

30. The Finney Procedure for Determining Convergence nth-Term Test Is ? No The series diverges. Converges to if . Diverges if . Yes or Maybe Geometric Series Test Is ? Yes No ?

31. Example Investigate the series to see if it diverges. Notice the series is NOT geometric. Using partial sums, it appears the series converges: Similar to a Geometric Series, we can investigate the ratio between terms. Similar to a converging Geometric Series, the limit of the “common” ratio appears to be less than 1. According to the next theorem. This means the series converges.

32. The Ratio Test Suppose the limit either exists or is infinite. Then: • If , the series converges. • If , the series diverges. • If , the test is inconclusive. Each successive terms are getting smaller. Each successive terms are getting larger. This is the “cousin” test to determining if a Convergent Geometric Series test. It is identical except: (1) If the “common” ratio is 1, then test fails. (2) The test does not determine the value the series converges to.

33. Example 1 Investigate the series to see if it diverges. Similar to a Geometric Series, we can investigate the ratio between terms. Similar to a converging Geometric Series, this “common” ratio is less than 1. According to the Ratio Test, this means the series converges.

34. Notice: Unlike the last example, this series depends on a value of x. Example 2 In #3 on Taylor Series Challenges we worked with the series . Find the radius of convergence. If the series converges, the ratio is less than 1. The interval of convergence is centered at 2 with a radius convergence of 3. The limit depends on n. Separate the n’s.

35. Example 3 Find the radius of convergence for the Maclaurin series for . We know: Thus: Investigate the ratio: The series has an infinite radius.

36. Example 4 Find the radius of convergence for the series . Investigate the ratio: Since the ratio tests uses an absolute value, the powers on -1 do not affect the limit. If the series converges, the ratio is less than 1. The interval of convergence is centered at 0 with a radius convergence of 5.

37. White Board Challenge Use the Ratio Test to determine the radius of convergence for . The Ratio Test is inconclusive because the limit is 1. We need to use other tests to determine if the series converges.

38. The Direct Comparison Test Suppose for all . • If converges, then converges. • If diverges, then diverges. Example: Prove convergesfor all real . Both series must have only positive terms. is the Taylor Series for . For all Since converges for all real , converges for all real by the Direct Comparison Test.

39. What about Negative Terms? Does the Direct Comparison Test fail if there are negative terms? Consider the series if : Using partial sums, it appears the series converges: But we don’t have a test to prove the series converges. Similar to the Ratio Test, what happens when we look at the absolute value of each term: Using partial sums, this series ALSO appears to converges: Since the new series is less than the original, if we can prove the new series converges, then the original series must converge.

40. Absolute Convergence Test If converges, then converges. Example: Prove converges for all . Unlike the Comparison Test, this test does not require the terms to be positive. converges by the Ratio Test: Notice: Investigate Since converges for all real by the Direct Comparison Test, converges for all real by the Absolute Convergence Test.

41. Definition of Absolute Convergence If converges, then converges absolutely. Example: The Alternating Harmonic Series converges but it does not converge absolutely: This is the divergent Harmonic Series.

42. Note about Absolute Convergence If converges, then converges absolutely. The Ratio Test uses an absolute value: . Thus, every test that converges by the Ratio Test also converges absolutely: converges absolutely because converges by by the ratio test The Ratio Test is incredibly strong