Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.

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1 0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate the sum of a alteratig series. Classify a coverget series as absolutely or coditioally coverget. Rearrage a ifiite series to obtai a differet sum. Alteratig Series So far, most series you have dealt with have had positive terms. I this sectio ad the followig sectio, you will study series that cotai both positive ad egative terms. The simplest such series is a alteratig series, whose terms alterate i sig. For example, the geometric series is a alteratig geometric series with r. Alteratig series occur i two ways: either the odd terms are egative or the eve terms are egative. THEOREM 9. Alteratig Series Test Let a > 0. The alteratig series a ad a coverge if the followig two coditios are met.. lim a 0. a a, for all Proof Cosider the alteratig series a. For this series, the partial sum (where is eve) S a a a a a 5 a... a a has all oegative terms, ad therefore S is a odecreasig sequece. But you ca also write S a a a a a 5... a a a which implies that S a for every iteger. So, S is a bouded, odecreasig sequece that coverges to some value L. Because S a S ad a 0, you have lim S lim S lim a L lim a L. Because both S ad S coverge to the same limit L, it follows that S also coverges to L. Cosequetly, the give alteratig series coverges. NOTE The secod coditio i the Alteratig Series Test ca be modified to require oly that 0 < a a for all greater tha some iteger N.

2 0_0905.qxd //0 :7 PM Page CHAPTER 9 Ifiite Series EXAMPLE Usig the Alteratig Series Test NOTE The series i Example is called the alteratig harmoic series more is said about this series i Example 7. Determie the covergece or divergece of Solutio Note that lim So, the first coditio of Theorem 9. is a lim 0. satisfied. Also ote that the secod coditio of Theorem 9. is satisfied because a a. for all. So, applyig the Alteratig Series Test, you ca coclude that the series coverges. EXAMPLE Usig the Alteratig Series Test Determie the covergece or divergece of Solutio To apply the Alteratig Series Test, ote that, for, So, a a for all. Furthermore, by L Hôpital s Rule, lim x x x lim x. x l 0. lim 0. Therefore, by the Alteratig Series Test, the series coverges. EXAMPLE Cases for Which the Alteratig Series Test Fails NOTE I Example (a), remember that wheever a series does ot pass the first coditio of the Alteratig Series Test, you ca use the th-term Test for Divergece to coclude that the series diverges. a. The alteratig series passes the secod coditio of the Alteratig Series Test because a a for all. You caot apply the Alteratig Series Test, however, because the series does ot pass the first coditio. I fact, the series diverges. b. The alteratig series... passes the first coditio because a approaches 0 as. You caot apply the Alteratig Series Test, however, because the series does ot pass the secod coditio. To coclude that the series diverges, you ca argue that S N equals the Nth partial sum of the diverget harmoic series. This implies that the sequece of partial sums diverges. So, the series diverges.

3 0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Alteratig Series Remaider For a coverget alteratig series, the partial sum S N ca be a useful approximatio for the sum S of the series. The error ivolved i usig S S N is the remaider R N S S N. THEOREM 9.5 Alteratig Series Remaider If a coverget alteratig series satisfies the coditio a a, the the absolute value of the remaider R N ivolved i approximatig the sum S by is less tha (or equal to) the first eglected term. That is, S S N R N a N. S N Proof The series obtaied by deletig the first N terms of the give series satisfies the coditios of the Alteratig Series Test ad has a sum of R N. R N S S N Cosequetly, a N a N a N N a N N a N... N a N a N a N... R N a N a N a N a N a N5... a N a N a N a N a N5... a N S S N R N a N, which establishes the theorem. EXAMPLE Approximatig the Sum of a Alteratig Series Approximate the sum of the followig series by its first six terms.!!!!! 5!!... TECHNOLOGY Later, i Sectio 9.0, you will be able to show that the series i Example coverges to e e 0.. For ow, try usig a computer to obtai a approximatio of the sum of the series. How may terms do you eed to obtai a approximatio that is withi uit of the actual sum? Solutio The series coverges by the Alteratig Series Test because!! ad The sum of the first six terms is S ad, by the Alteratig Series Remaider, you have S S R a So, the sum S lies betwee ad , ad you have 0.7 S 0.. lim 0.!

4 0_0905.qxd //0 :7 PM Page CHAPTER 9 Ifiite Series Absolute ad Coditioal Covergece Occasioally, a series may have both positive ad egative terms ad ot be a alteratig series. For istace, the series si si si si 9... has both positive ad egative terms, yet it is ot a alteratig series. Oe way to obtai some iformatio about the covergece of this series is to ivestigate the covergece of the series si. By direct compariso, you have si for all, so si., Therefore, by the Direct Compariso Test, the series si coverges. The ext theorem tells you that the origial series also coverges. THEOREM 9. Absolute Covergece If the series coverges, the the series a also coverges. Proof Because for all, the series a a coverges by compariso with the coverget series a. a Furthermore, because 0 a a a a a a a, a a a a you ca write where both series o the right coverge. So, it follows that a coverges. The coverse of Theorem 9. is ot true. For istace, the alteratig harmoic series... coverges by the Alteratig Series Test. Yet the harmoic series diverges. This type of covergece is called coditioal. Defiitios of Absolute ad Coditioal Covergece. a is absolutely coverget if coverges.. a is coditioally coverget if a coverges but diverges. a a

5 0_0905.qxd //0 :7 PM Page 5 SECTION 9.5 Alteratig Series 5 EXAMPLE 5 Absolute ad Coditioal Covergece Determie whether each of the series is coverget or diverget. Classify ay coverget series as absolutely or coditioally coverget. a. b.! 0!!!! Solutio a. By the th-term Test for Divergece, you ca coclude that this series diverges. b. The give series ca be show to be coverget by the Alteratig Series Test. Moreover, because the p-series... diverges, the give series is coditioally coverget. EXAMPLE Absolute ad Coditioal Covergece Determie whether each of the series is coverget or diverget. Classify ay coverget series as absolutely or coditioally coverget. a. b l l l l l 5... Solutio a. This is ot a alteratig series. However, because ( is a coverget geometric series, you ca apply Theorem 9. to coclude that the give series is absolutely coverget (ad therefore coverget). b. I this case, the Alteratig Series Test idicates that the give series coverges. However, the series l l l l... diverges by direct compariso with the terms of the harmoic series. Therefore, the give series is coditioally coverget. Rearragemet of Series A fiite sum such as 5 ca be rearraged without chagig the value of the sum. This is ot ecessarily true of a ifiite series it depeds o whether the series is absolutely coverget (every rearragemet has the same sum) or coditioally coverget.

6 0_0905.qxd //0 :7 PM Page CHAPTER 9 Ifiite Series EXAMPLE 7 Rearragemet of a Series FOR FURTHER INFORMATION Georg Friedrich Riema (8 8) proved that if a is coditioally coverget ad S is ay real umber, the terms of the series ca be rearraged to coverge to S. For more o this topic, see the article Riema s Rearragemet Theorem by Stewart Galaor i Mathematics Teacher. To view this article, go to the website The alteratig harmoic series coverges to l. That is,... l. Rearrage the series to produce a differet sum. Solutio Cosider the followig rearragemet (See Exercise 9, Sectio 9.0.) l By rearragig the terms, you obtai a sum that is half the origial sum. I Exercises, match the series with the graph of its sequece of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), ad (f).] (a) (c) (e) Exercises for Sectio (b) (d) (f)...!. S S S S S S 8 0! Numerical ad Graphical Aalysis the Alteratig Series Remaider. I Exercises 7 0, explore (a) Use a graphig utility to fid the idicated partial sum ad complete the table. (b) Use a graphig utility to graph the first 0 terms of the sequece of partial sums ad a horizotal lie represetig the sum. (c) What patter exists betwee the plot of the successive poits i part (b) relative to the horizotal lie represetig the sum of the series? Do the distaces betwee the successive poits ad the horizotal lie icrease or decrease? (d) Discuss the relatioship betwee the aswers i part (c) ad the Alteratig Series Remaider as give i Theorem ! e 0. See for worked-out solutios to odd-umbered exercises. si! S S

7 0_0905.qxd //0 :7 PM Page 7 SECTION 9.5 Alteratig Series 7 I Exercises, determie the covergece or divergece of the series..... l l l.. si si.. cos cos 5.. 0! 0! ! 0... I Exercises, approximate the sum of the series by usig the first six terms. (See Example.).. l 5.. 0! I Exercises 7, (a) use Theorem 9.5 to determie the umber of terms required to approximate the sum of the coverget series with a error of less tha 0.00, ad (b) use a graphig utility to approximate the sum of the series with a error of less tha ! e si 0! 0. cos e e e e! l l 5 csch sech 0! e I Exercises, use Theorem 9.5 to determie the umber of terms required to approximate the sum of the series with a error of less tha I Exercises 7, determie whether the series coverges coditioally or absolutely, or diverges l ! arcta cos cos si Writig About Cocepts e Defie a alteratig series ad state the Alteratig Series Test.. Give the remaider after N terms of a coverget alteratig series. 5. I your ow words, state the differece betwee absolute ad coditioal covergece of a alteratig series.. The graphs of the sequeces of partial sums of two series are show i the figures. Which graph represets the partial sums of a alteratig series? Explai. (a) (b) S S

8 0_0905.qxd //0 :7 PM Page 8 8 CHAPTER 9 Ifiite Series True or False? I Exercises 7 70, determie whether the statemet is true or false. If it is false, explai why or give a example that shows it is false. 7. If both a ad a coverge, the coverges. 8. If a diverges, the a diverges. 9. For the alteratig series the partial sum S 00 is a, overestimate of the sum of the series. 70. If a ad b both coverge, the a b coverges. I Exercises 7 ad 7, fid the values of p for which the series coverges Prove that if coverges, the a coverges. Is the coverse true? If ot, give a example that shows it is false. 7. Use the result of Exercise 7 to give a example of a alteratig p-series that coverges, but whose correspodig p-series diverges. 75. Give a example of a series that demostrates the statemet you proved i Exercise Fid all values of x for which the series x (a) coverges absolutely ad (b) coverges coditioally. 77. Cosider the followig series. (a) Does the series meet the coditios of Theorem 9.? Explai why or why ot. (b) Does the series coverge? If so, what is the sum? 78. Cosider the followig series.,, (a) Does the series meet the coditios of Theorem 9.? Explai why or why ot. (b) Does the series coverge? If so, what is the sum? Review I Exercises 79 88, test for covergece or divergece ad idetify the test used p a, a 8 00e a if is odd if is eve a p l 89. The followig argumet, that 0, is icorrect. Describe the error The followig argumet,, is icorrect. Describe the error. Multiply each side of the alteratig harmoic series S 5 7 by to get S Now collect terms with like deomiators (as idicated by the arrows) to get S The resultig series is the same oe that you started with. So, S S ad divide each side by S to get. FOR FURTHER INFORMATION For more o this exercise, see the article Riema s Rearragemet Theorem by Stewart Galaor i Mathematics Teacher. To view this article, go to the website Putam Exam Challege 9. Assume as kow the (true) fact that the alteratig harmoic series () is coverget, ad deote its sum by s. Rearrage the series () as follows: () Assume as kow the (true) fact that the series () is also coverget, ad deote its sum by S. Deote by s k, S k the kth partial sum of the series () ad (), respectively. Prove each statemet. (i) S s s, (ii) S s This problem was composed by the Committee o the Putam Prize Competitio. The Mathematical Associatio of America. All rights reserved.