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11.6 Absolute Convergence and the Ratio and Root tests

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. ABSOLUTE CONVERGENCE. Given any series  a n , we can consider the corresponding series

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11.6 Absolute Convergence and the Ratio and Root tests

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  1. 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.

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

  3. ABSOLUTE CONVERGENCE Definition 1 • A series an is called absolutely convergentif the series of absolute values  |an| is convergent. • Notice that, if an is a series with positive terms, then |an| = an. • So, in this case,absolute convergence is the same as convergence.

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

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

  6. 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.

  7. CONDITIONAL CONVERGENCE Definition 2 • A series anis called conditionally convergentif it is convergent but not absolutely convergent.

  8. 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.

  9. ABSOLUTE CONVERGENCE Theorem 3 • If a series anis absolutely convergent, then it is convergent.

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

  11. ABSOLUTE CONVERGENCE Theorem 3 - Proof • If an is absolutely convergent, then  |an|is convergent. • So,  2|an|is convergent. • Thus, by the Comparison Test,  (an+ |an|) is convergent.

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

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

  14. 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.

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

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

  17. ABSOLUTE CONVERGENCE Example 3 • Thus, the given series  (cos n)/n2 is absolutely convergent and therefore, convergent by Theorem 3.

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

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

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

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

  22. 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 r such that L < r < 1.

  23. 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:

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

  25. 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

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

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

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

  29. THE RATIO TEST Case i - Proof • It follows that the series is convergent. • Recall that a finite number of terms does not affect convergence. • Therefore, anis absolutely convergent.

  30. 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:

  31. THE RATIO TEST Case ii - Proof • This means that |an+1| > |an| whenever nN, and so • Therefore, andiverges by the Test for Divergence.

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

  33. NOTE Case iii - Proof • For instance, for the convergent series  1/n2, we have:

  34. NOTE Case iii - Proof • For the divergent series  1/n, we have:

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

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

  37. RATIO TEST Example 4

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

  39. RATIO TEST Example 5 • Test the convergence of the series • Since the terms an = nn/n! are positive, we do not need the absolute value signs.

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

  41. 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.

  42. 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.

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

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

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

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

  47. ROOT TEST VS. RATIO TEST • If L = 1 in the Ratio Test, do not try the Root Test, because L will again be 1. • If L = 1 in the Root Test, do not try the Ratio Test, because it will fail too.

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

  49. REARRANGEMENTS • The question of whether a given convergent series • is absolutely convergent or conditionally convergent has a bearing on the question of whether infinite sums behave like finite sums.

  50. REARRANGEMENTS • If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged. • However, this is not always the case for an infinite series.

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