INFINITE SEQUENCES AND SERIES

1. 11 INFINITE SEQUENCES AND SERIES

2. INFINITE SEQUENCES AND SERIES 11.6 Absolute Convergence and the Ratio and Root tests In this section, we will learn about: Absolute convergence of a series and tests to determine it.

3. ABSOLUTE CONVERGENCE • Given any series Σan, we can consider the corresponding series • whose terms are the absolute values of the terms of the original series.

4. ABSOLUTE CONVERGENCE Definition 1 • A series Σan is called absolutely convergentif the series of absolute values Σ |an| is convergent.

5. ABSOLUTE CONVERGENCE • Notice that, if Σan is a series with positive terms, then |an| = an. • So, in this case,absolute convergence is the same as convergence.

6. ABSOLUTE CONVERGENCE Example 1 • The series • is absolutely convergent because • is a convergent p-series (p = 2).

7. ABSOLUTE CONVERGENCE Example 2 • We know that the alternating harmonic series • is convergent. • See Example 1 in Section 11.5.

8. ABSOLUTE CONVERGENCE Example 2 • However, it is not absolutely convergent because the corresponding series of absolute values is: • This is the harmonic series (p-series with p = 1) and is, therefore, divergent.

9. CONDITIONAL CONVERGENCE Definition 2 • A series Σanis called conditionally convergentif it is convergent but not absolutely convergent.

10. ABSOLUTE CONVERGENCE • Example 2 shows that the alternating harmonic series is conditionally convergent. • Thus, it is possible for a series to be convergent but not absolutely convergent. • However, the next theorem shows that absolute convergence implies convergence.

11. ABSOLUTE CONVERGENCE Theorem 3 • If a series Σanis absolutely convergent, then it is convergent.

12. ABSOLUTE CONVERGENCE Theorem 3—Proof • Observe that the inequality • is true because |an| is either an or –an.

13. ABSOLUTE CONVERGENCE Theorem 3—Proof • If Σan is absolutely convergent, then Σ |an|is convergent. • So, Σ 2|an|is convergent. • Thus, by the Comparison Test, Σ (an+ |an|) is convergent.

14. ABSOLUTE CONVERGENCE Theorem 3—Proof • Then, • is the difference of two convergent series and is, therefore, convergent.

15. ABSOLUTE CONVERGENCE Example 3 • Determine whether the series • is convergent or divergent.

16. ABSOLUTE CONVERGENCE Example 3 • The series has both positive and negative terms, but it is not alternating. • The first term is positive. • The next three are negative. • The following three are positive—the signs change irregularly.

17. ABSOLUTE CONVERGENCE Example 3 • We can apply the Comparison Test to the series of absolute values:

18. ABSOLUTE CONVERGENCE Example 3 • Since |cos n| ≤ 1 for all n, we have: • We know that Σ 1/n2 is convergent (p-series with p = 2). • Hence, Σ (cos n)/n2 is convergent by the Comparison Test.

19. ABSOLUTE CONVERGENCE Example 3 • Thus, the given series Σ (cos n)/n2is absolutely convergent and, therefore, convergent by Theorem 3.

20. ABSOLUTE CONVERGENCE • The following test is very useful in determining whether a given series is absolutely convergent.

21. THE RATIO TEST Case i • If • then the series is absolutely convergent (and therefore convergent).

22. THE RATIO TEST Case ii • If • then the series is divergent.

23. THE RATIO TEST Case iii • If • the Ratio Test is inconclusive. • That is, no conclusion can be drawn about the convergence or divergence of Σan.

24. THE RATIO TEST Case i—Proof • The idea is to compare the given series with a convergent geometric series. • Since L < 1, we can choose a number rsuch that L < r < 1.

25. THE RATIO TEST Case i—Proof • Since • the ratio |an+1/an| will eventually be less than r. • That is, there exists an integer N such that:

26. THE RATIO TEST i-Proof (Inequality 4) • Equivalently, • |an+1| < |an|r whenever n≥ N

27. THE RATIO TEST Case i—Proof • Putting n successively equal to N, N + 1, N + 2, . . . in Equation 4, we obtain: • |aN+1| < |aN|r • |aN+2| < |aN+1|r < |aN|r2 • |aN+3| < |aN+2| < |aN|r3

28. THE RATIO TEST i-Proof (Inequality 5) • In general, • |aN+k| < |aN|rk for all k≥ 1

29. THE RATIO TEST Case i—Proof • Now, the series • is convergent because it is a geometric series with 0 < r < 1.

30. THE RATIO TEST Case i—Proof • Thus, the inequality 5, together with the Comparison Test, shows that the series • is also convergent.

31. THE RATIO TEST Case i—Proof • It follows that the series is convergent. • Recall that a finite number of terms doesn’t affect convergence. • Therefore, Σanis absolutely convergent.

32. THE RATIO TEST Case ii—Proof • If |an+1/an| →L > 1 or |an+1/an| →∞ • then the ratio |an+1/an|will eventually be greater than 1. • That is, there exists an integer N such that:

33. THE RATIO TEST Case ii—Proof • This means that |an+1| > |an| whenever n≥ N, and so • Therefore, Σandiverges by the Test for Divergence.

34. NOTE Case iii—Proof • Part iii of the Ratio Test says that, if • the test gives no information.

35. NOTE Case iii—Proof • For instance, for the convergent series Σ 1/n2, we have:

36. NOTE Case iii—Proof • For the divergent series Σ 1/n, we have:

37. NOTE Case iii—Proof • Therefore, if , the series Σanmight converge or it might diverge. • In this case, the Ratio Test fails. • We must use some other test.

38. RATIO TEST Example 4 • Test the series for absolute convergence. • We use the Ratio Test with an = (–1)n n3 / 3n,asfollows.

39. RATIO TEST Example 4

40. RATIO TEST Example 4 • Thus, by the Ratio Test, the given series is absolutely convergent and, therefore, convergent.

41. RATIO TEST Example 5 • Test the convergence of the series • Since the terms an = nn/n! are positive, we don’t need the absolute value signs.

42. RATIO TEST Example 5 • See Equation 6 in Section 3.6 • Since e > 1, the series is divergent by the Ratio Test.

43. NOTE • Although the Ratio Test works in Example 5, an easier method is to use the Test for Divergence. • Since it follows that an does not approach 0 as n→ ∞. • Thus, the series is divergent by the Test for Divergence.

44. ABSOLUTE CONVERGENCE • The following test is convenient to apply when nth powers occur. • Its proof is similar to the proof of the Ratio Test and is left as Exercise 37.

45. THE ROOT TEST Case i • If • then the series is absolutely convergent (and therefore convergent).

46. THE ROOT TEST Case ii • If • then the series is divergent.

47. THE ROOT TEST Case iii • If • the Root Test is inconclusive.

48. ROOT TEST • If , then part iii of the Root Test says that the test gives no information. • The series Σancould converge or diverge.

49. ROOT TEST VS. RATIO TEST • If L = 1 in the Ratio Test, don’t try the Root Test—because L will again be 1. • If L = 1 in the Root Test, don’t try the Ratio Test—because it will fail too.

50. ROOT TEST Example 6 • Test the convergence of the series • Thus, the series converges by the Root Test.