Testing Series with Mixed Terms

Transcription

1 Testing Series with Mixed Terms Philippe B. Laval KSU Today Philippe B. Laval (KSU) Series with Mixed Terms Today 1 / 17

2 Outline 1 Introduction 2 Absolute v.s. Conditional Convergence 3 Alternating Series 4 The Ratio and Root Tests 5 Conclusion Philippe B. Laval (KSU) Series with Mixed Terms Today 2 / 17

3 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. Philippe B. Laval (KSU) Series with Mixed Terms Today 3 / 17

4 Absolute v.s. Conditional Convergence Given a series Σa n = a 1 + a 2 + a , consider the series of absolute values: Σ a n = a 1 + a 2 + a Which one is more likely to converge, why, what approach to testing series with mixed terms does it suggest? Remark Obviously, if a n is a series of positive terms, then a n = a n. We will use this obvious fact below. In working with absolute values, it is important to remember some of its properties we list here for convenience. Let a, b and c denote real numbers and n a positive integer. Then: ab = a b a n = a n a < c c < a < c a = a b b If a 0 then a = a If a < 0 then a = a Philippe B. Laval (KSU) Series with Mixed Terms Today 4 / 17

5 Absolute v.s. Conditional Convergence What is a n if a n = ( 1)n n! Definition (Absolute Convergence) and if a n = ( 1)n x n? n! A series a n is said to be absolutely convergent if the series of absolute values ( a n = a 1 + a 2 + a ) is convergent. Definition (Conditional Convergence)) A series a n is said to be conditionally convergent if it converges but the series of absolute values ( a n ) diverges. Theorem If a series converges absolutely, then it also converges in other words if Σ a n converges then Σa n also converges. Philippe B. Laval (KSU) Series with Mixed Terms Today 5 / 17

6 Absolute v.s. Conditional Convergence Remark Let us make a few remarks: 1 It now seems that we have two types of convergence? What is the difference? 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. 3 For a series of positive terms, both notions are the same. 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. Philippe B. Laval (KSU) Series with Mixed Terms Today 6 / 17

7 If looking Philippe B. at Laval the (KSU) series of absolute Series with values Mixed Terms does not provide an answer, Today we7 / 17 Absolute v.s. Conditional Convergence ( 1) n 1 Is n divergent? ( 1) n 1 Is divergent? n 2 absolutely convergent, conditionally convergent or absolutely convergent, conditionally convergent or ( 1) n 3n Is 4n 1 divergent? absolutely convergent, conditionally convergent or

8 Alternating Series Definition (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 > is an alternating series is an alternating series. Philippe B. Laval (KSU) Series with Mixed Terms Today 8 / 17

9 Alternating Series Theorem (Alternating Series Test) If ( 1) n 1 b n, where b n > 0 satisfies: 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. Determine if the alternating ( 1) n 1 harmonic series n converges. Determine if converges. ( 1) n 3n 4n 1 Philippe B. Laval (KSU) Series with Mixed Terms Today 9 / 17

10 Alternating Series Theorem (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 + 1 term. ( 1) n 1 Find the sum of n! with an error less than.001. The next two tests are extremely important, especially the first one. It is the most widely used test. Philippe B. Laval (KSU) Series with Mixed Terms Today 10 / 17

11 The Ratio and Root Tests Theorem (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 The ratio test works best with series which involve factorials and other products. Philippe B. Laval (KSU) Series with Mixed Terms Today 11 / 17

12 The Ratio and Root Tests Theorem (Root Test) Let a n be a series and suppose that L = lim 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. n n an exists or is infinite Remark If L = 1 in the ratio test, do not try the root test, L will also be 1. Remark The root test works with series whose general term is a power of n. Philippe B. Laval (KSU) Series with Mixed Terms Today 12 / 17

13 The Ratio and Root Tests ( 1) n n 3 Test 3 n. 2 n Test n! 1 Test (ln n) n. n=2 Philippe B. Laval (KSU) Series with Mixed Terms Today 13 / 17

14 The Ratio and Root Tests Find x so that Find x so that Find x so that n=0 x n n! converges. 2 n x n converges. n=0 ( 1) n (x 1) n converges. n Philippe B. Laval (KSU) Series with Mixed Terms Today 14 / 17

15 Conclusion When testing a series, proceed in the following order. 1 If the series is a known series you know exactly what to do. 2 Use the test for divergence. If lim a n 0, then a n will diverge. n 3 If the series is an alternating series, use the alternating series test. 4 For other series with negative terms look at the series of absolute values. If it converges, the original series also will. 5 Series which involve factorials and other products (but not rational functions in n) should be tested with the ratio test. 6 Series for which the general term is a power of n (but not rational functions in n) should be tested with the root test. 7 Series with positive terms, 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. Philippe B. Laval (KSU) Series with Mixed Terms Today 15 / 17 i=1

16 Practice Using the outline above, decide which test you would use on the given series. If they are known series, identify them. 2 n n n n=0 n=2 n=2 4 n+1 5 n 1 n ln n 2n + 1 3n n 2 1 ( 1) n n n n=2 ( 3) n n 3 n (2n)! ( 1) n 1 1 n + 4 Philippe B. Laval (KSU) Series with Mixed Terms Today 16 / 17

17 Exercises See the problems at the end of my notes on series with mixed terms. Philippe B. Laval (KSU) Series with Mixed Terms Today 17 / 17