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Copyright © Cengage Learning. All rights reserved. 11 Infinite Sequences and Series
Copyright © Cengage Learning. All rights reserved. 11.6 Absolute Convergence and the Ratio and Root Tests
3 Given any series a n, we can consider the corresponding series whose terms are the absolute values of the terms of the original series.
4 Absolute Convergence and the Ratio and Root Tests Notice that if a n is a series with positive terms, then | a n | = a n and so absolute convergence is the same as convergence in this case.
5 Example 1 The series is absolutely convergent because is a convergent p-series (p = 2).
6 Example 2 We know that the alternating harmonic series is convergent, but it is not absolutely convergent because the corresponding series of absolute values is which is the harmonic series (p-series with p = 1) and is therefore divergent.
7 Absolute Convergence and the Ratio and Root Tests 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.
8 Example 3 Determine whether the series is convergent or divergent. Solution: This series has both positive and negative terms, but it is not alternating. (The first term is positive, the next three are negative, and the following three are positive: The signs change irregularly.)
9 Example 3 – Solution We can apply the Comparison Test to the series of absolute values Since | cos n | 1 for all n, we have We know that 1/n 2 is convergent (p-series with p = 2) and therefore | cos n |/n 2 is convergent by the Comparison Test. cont’d
10 Example 3 – Solution Thus the given series (cos n)/n 2 is absolutely convergent and therefore convergent by Theorem 3. cont’d
11 Absolute Convergence and the Ratio and Root Tests The following test is very useful in determining whether a given series is absolutely convergent.
12 Absolute Convergence and the Ratio and Root Tests Note: Part (iii) of the Ratio Test says that if the test gives no information. For instance, for the convergent series 1/n 2 we have
13 Absolute Convergence and the Ratio and Root Tests whereas for the divergent series 1/n we have Therefore, if the series a n might converge or it might diverge. In this case the Ratio Test fails and we must use some other test.
14 Example 5 Test the convergence of the series Solution: Since the terms a n = n n /n! are positive, we don’t need the absolute value signs.
15 Example 5 – Solution as n Since e > 1, the given series is divergent by the Ratio Test. cont’d
16 Absolute Convergence and the Ratio and Root Tests Note: Although the Ratio Test works in Example 5, an easier method is to use the Test for Divergence. Since it follows that a n does not approach 0 as n. Therefore the given series is divergent by the Test for Divergence.
17 Absolute Convergence and the Ratio and Root Tests The following test is convenient to apply when n th powers occur.
18 Absolute Convergence and the Ratio and Root Tests If then part (iii) of the Root Test says that the test gives no information. The series a n could converge or diverge. (If L = 1 in the Ratio Test, don’t try the Root Test because L will again be 1. And if L = 1 in the Root Test, don’t try the Ratio Test because it will fail too.)
19 Example 6 Test the convergence of the series Solution: Thus the given series converges by the Root Test.
21 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. If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged. But this is not always the case for an infinite series.
22 Rearrangements By a rearrangement of an infinite series a n we mean a series obtained by simply changing the order of the terms. For instance, a rearrangement of a n could start as follows: a 1 + a 2 + a 5 + a 3 + a 4 + a 15 + a 6 + a 7 + a 20 +… It turns out that if a n is absolutely convergent series with sum s, then any rearrangement of a n has the same sum s.
23 Rearrangements However, any conditionally convergent series can be rearranged to give a different sum. To illustrate this fact let’s consider the alternating harmonic series If we multiply this series by we get
24 Rearrangements Inserting zeros between the terms of this series, we have Now we add the series in Equations 6 and 7:
25 Rearrangements Notice that the series in contains the same terms as in but rearranged so that one negative term occurs after each pair of positive terms. The sums of these series, however, are different. In fact, Riemann proved that if a n is a conditionally convergent series and r is any real number whatsoever, then there is a rearrangement of a n that has a sum equal to r.
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