Testing Series With Mixed Terms

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1 Testing Series With Mixed Terms Philippe B. Laval Series with Mixed Terms 1. Introduction 2. Absolute v.s. Conditional Convergence 3. Alternating Series 4. The Ratio and Root Tests 5. Conclusion 1

2 Introduction Except for geometric series, the tests we have developed so far only applied to series with positive terms. How do we handle series with mixed terms? From today on, unless stated otherwise, series will have mixed terms. 2

3 Absolute v.s Conditional Convergence Given a series Σa n = a 1 + a 2 + a , consider the series Σ a n = a 1 + a 2 + a Which one is more likely to converge and why? Theorem 1 If Σ a n converges then Σa n also converges. Definition 1 (Absolute Convergence) Aseries an is said to be absolutely convergent if the series of absolute values ( a n = a 1 + a 2 + a ) is convergent. Definition 2 (Conditional Convergence)) Aseries a n is said to be conditionally convergent if it converges but the series of absolute values ( a n = a 1 + a 2 + a ) diverges. The above theorem can be restated as follows: 3

4 Theorem 2 If a series converges absolutely, then it also converges. 4

5 Absolute v.s Conditional Convergence Remark 1 It now seems that we have two types of convergence? What is the difference? Remark 2 Both types of convergence imply that the series (infinite sum) exists and is finite. In fact, for this class, you will not use the extra properties that absolute convergence has. Remark 3 For a series of positive terms, both notions are the same. Remark 4 The theorem provides a way to study the convergence of series with mixed terms. If we have to study the convergence of Σa n. We can look at Σ a n. If it converges, then it means that Σa n converges absolutely and thus converges. 5

6 Absolute v.s Conditional Convergence ( 1) n 1 Example 1 Is absolutely convergent, conditionally convergent or n divergent? n 2 ( 1) n 1 Example 2 Is absolutely convergent, conditionally convergent or divergent? ( 1) n 3n Example 3 Is absolutely convergent, conditionally convergent or 4n 1 divergent? If looking at the series of absolute values does not provide an answer, we look at new tests. The first test we develop applies to series where the posite and negative terms alternate. 6

7 Alternating Series Definition 3 (Alternating Series) An alternating series is a series whose terms are alternatively positive and negative. We usually write an alternating series as ( 1) n 1 b n = b 1 b 2 + b 3 b 4... or ( 1) n b n = b 1 + b 2 b 3... where b n > 0. Example is an alternating 4 series. Example is an alternating series. 7

8 Alternating Series Theorem 3 (Alternating Series Test) If where b n > 0 satisfies: ( 1) n 1 b n, 1. b n+1 b n for all n from some point on, and 2. lim n b n =0 then the series is convergent. ( 1) n 1 Example 6 Determine if the series, n known as the alternating harmonic series, converges. Example 7 Determine if ( 1) n 3n 4n 1 converges. 8

9 Alternating Series Theorem 4 (Approximating Alternating Series) If S = ( 1) n 1 b n and b n satisfies the conditions of the alternating series test, then R n = S S n b n+1 In other words, the error by approximating the sum of a convergent alternating series by the sum of the first n terms is no greater than the n +1term. Example 8 Find the sum of an error less than.001. n=0 ( 1) n n! with The next two tests are extremey important, especially the first one. It is the most widely used test. 9

10 Ratios and Root Tests Theorem 5 (Ratio Test) Let a n be a series of non-zero terms and suppose that L = lim a n+1 n a n exists or is infinite then: 1. If L<1, a n converges absolutely 2. If L>1, a n diverges 3. If L =1, the test provides no conclusion. Remark 5 The ratio test works best with series which involve factorials and other products. 10

11 Ratios and Root Tests Theorem 6 (Root Test) Let a n be a series and suppose that L = lim n a n exists or is n infinite then: 1. If L<1, a n converges absolutely 2. If L>1, a n diverges 3. If L =1, the test provides no conclusion. Remark 6 If L =1in the ratio test, do not try the root test, L will also be 1. Remark 7 The root test works with series whose general term is a power of n. 11

12 Ratios and Root Tests Example 9 Test ( 1) n n 3 3 n. Example 10 Test 2 n n! Example 11 Test n=2 1 (ln n) n. Example 12 Find x so that n=0 x n n! converges. Example 13 Find x so that 2 n x n converges. n=0 12

13 Example 14 Find x so that converges. n=0 ( 1) n n (x 1)n 13

14 Conclusion When testing a series, proceed in the following order. 1. See if the series is a known series, such as a p-series or a geometric series, or a harmonic series,... In this case, you know exactly what to do. 2. Before applying a test, check that it is worth it, i.e. can the series converge? This is done by applying the test for divergence. If lim a n 0,then a n will diverge. n i=1 3. If the series has negative terms, and is an alternating series, use the alternating series test. 4. If the series has negative terms, and is not an alternating series, look at the series of absolute values. If it converges, the original series also will. 5. Series which involve factorials and other products should be tested with the ratio 14

15 test. 6. Series for which the general term is a power of n should be tested with the root test. 7. Series which are similar to a p-series or a geometric series should be compared to a p-series or a geometric series. 8. Series for which the general term is a function which can be easily integrated should be tested with the integral test. 15

16 Practice Using the outline above, decide which test you would use on the given series. If they are known series, identify them. 4 n n=0 n=2 5 n 1 n ln n 2n +1 3n +1 1 n n n n ( 1) n n=2 ( 3) n n n n

17 n (2n)! ( 1) n 1 1 n +4 17

18 Homework Do #1, 3, 5, 7, 9, 13, 17, 19, 21, 23, 25, 31, 33 on pages 598,

Testing Series with Mixed Terms

Testing Series with Mixed Terms Philippe B. Laval KSU Today Philippe B. Laval (KSU) Series with Mixed Terms Today 1 / 17 Outline 1 Introduction 2 Absolute v.s. Conditional Convergence 3 Alternating Series