Roberto s Notes on Series Chapter 2: Convergence tests Section 7. Alternating series

Transcription

1 Roberto s Notes o Series Chapter 2: Covergece tests Sectio 7 Alteratig series What you eed to kow already: All basic covergece tests for evetually positive series. What you ca lear here: A test for series whose terms alterate i sig. The covergece tests we have see so far require the series to be evetually positive. Of course, by factorig out a egative, they ca be made to work also for series that are evetually egative. But what about series that have ifiitely may positive ad ifiitely may egative terms? It turs out that thigs get really iterestig for such series, sometimes i the sese of really simple covergece criteria, sometimes by creatig some couterituitive ad strage situatios. Oe situatio where thigs that ca be hadled very easily ad efficietly relates to a special type of series of this kid. A series 1 Defiitio a is said to be a alteratig series if its terms alterate i their sig. Formally this meas that oe of these two coditios hold: For all values of, a 1 For all values of, 1 a 1 a a, or Alteratively, a alteratig series is obtaied by startig from a positive sequece a ad cosiderig oe of the two series: a or a 1 1 Notice that the two series associated to the same positive sequece are opposites of each other. Example: This is a alteratig series geerated by the harmoic sequece a, ad it is therefore called the alteratig harmoic series. Ad so is the opposite series: Series Chapter 2: Covergece tests Sectio 7: Alteratig series Page 1

2 I hope you remember that the harmoic series diverges. I also hope you remember that the divergece test oly works i oe directio, that is, it ca prove divergece, but ot covergece. Well, the followig, very importat fact about alteratig series shows that for them thigs are differet ad icer. Techical fact A alteratig series a 1 is coverget if ad oly if the positive sequece a is evetually decreasig ad lim a 0. Strategy for estimatig the sum of a coverget alteratig series The sum of a coverget alteratig series a 1 differs from its -th partial sum by less tha the absolute value of a 1: S S a 1 Therefore, such sum is bouded by ay two cosecutive partial sums: S S S if a S S S 1 if a 1 0 This meas that for alteratig series, the divergece test may be reversed to provide a covergece test, as log as the sequece is also decreasig. I will spare you the formal details of the proof ad focus istead o its essece. Outlie of the proof Sice a is positive ad decreasig to 0, each partial sum is obtaied from the previous oe by alteratively addig ad subtractig a umber that gets smaller ad smaller. This meas that the terms of the sequece of partial sums become bouded withi smaller ad smaller itervals, evetually beig squeezed dow to a sigle value, amely its limit. The idea highlighted i this outlie also suggests a way to estimate the sum of a coverget alteratig series. Example: If we compute the 10-th partial sum of this series we get: S Therefore, at this poit we kow that: 1 this value is closer to the true sum by less tha , hece: the true sum is S Series Chapter 2: Covergece tests Sectio 7: Alteratig series Page 2

3 It turs out that the true value of the sum of the alteratig harmoic series is l 2, which is approximately , but we eed more kowledge about series to prove that! These two properties of alteratig series cast them uder a favourable light: it is easy to check if they coverge ad we ca estimate their sum fairly easily. But there is a dark side to them! For some coverget alteratig series the terms ca be rearraged so that the ew series will coverge to ay real umber, or eve diverge. I other words, some such series behave i a very bizarre way uder rearragemets ad we must be careful i their hadlig. We shall see soo how to idetify the alteratig series that belog to this rebellious group. Before movig to the practice questios, here is somethig to keep i mid. Kot o your figer Although the alteratio i a alteratig series is traditioally deoted by the factor 1, it ca be represeted i other ways as well. I particular, the expressio cos also provides alteratig values of 1 ad -1 ad it has the advatage of startig o a positive ote. Example: 3 cos ta I this series, the cos factor tells us that the series is alteratig, while the ta factor approaches 0 as approaches. Therefore, by the alteratig series test this series is coverget. Notice that we had to start the series at 3 to esure that all terms would be defied, but this does ot affect the validity of the test or of the approximatio. For istace we ca quickly see that, sice cos5 1: ta ta ta cos ta ta ta cos ta This is ot a great approximatio for such a small umber, so we may wat to use more terms to get a more decet estimate. Ulike the case for the alteratig harmoic series, it is ot easy to fid what the series of the last example coverges to. I fact I do ot thik that such limitig value is kow! There are may other series whose limit is ot kow, but we shall soo see a method to fid limitig values i may situatios, most of which stem from alteratig series. Series Chapter 2: Covergece tests Sectio 7: Alteratig series Page 3

4 Summary A alteratig series is a series whose terms alterate betwee beig positive ad beig egative. For a alteratig series the divergece test ca be iverted to provide a simple covergece test. Commo errors to avoid Just because a series has both positive ad egative terms, it does ot mea that it is alteratig. To be alteratig, the sig of the terms must alterate from each term to the ext. Learig questios for Sectio S 2-7 Review questios: 1. Explai what a alteratig series is. 2. Describe the mai covergece criterio for a alteratig series. 3. Explai why the estimatio formula for the sum of a coverget alteratig series works. Memory questios: 1. What is a alteratig series? 3. Which formula approximates the sum of a coverget alteratig series? 2. State the alteratig series test. Series Chapter 2: Covergece tests Sectio 7: Alteratig series Page 4

5 Computatio questios: For each of the series provided i questios 1-16: Determie if it is alteratig. If it is evetually alteratig, determie if it is coverget. If it is coverget, estimate its sum to 2 decimal digits l cos l ! cos ! 1 1 2! / cos si cos si cos 1! si 2 cos ! 1 1!cos Series Chapter 2: Covergece tests Sectio 7: Alteratig series Page 5

6 17. Which is the shortest partial sum of the series 0 1! that provides a estimate of its sum that is closer tha to the actual sum? 18. For what value of p is the series cos coverget? 2 p 1 Theory questios: 1. What is the mai good feature of alteratig series? 2. What is the mai bad feature of alteratig series? 3. What does the alteratig harmoic series coverge to? Read the sectio carefully! 4. Give a example of a diverget alteratig series What questios do you have for your istructor? Series Chapter 2: Covergece tests Sectio 7: Alteratig series Page 6