2.5. THE COMPARISON TEST

Transcription

1 SECTION. THE COMPARISON TEST 69.. THE COMPARISON TEST We bega our systematic study of series with geometric series, provig the Geometric Series Test: ar coverges if ad oly if r. The i the last sectio we compared series to itegrals i order to determie if they coverge or diverge, ad established the p-series Test: p coverges if ad oly if p. I this sectio we study aother type of compariso where we compare series to other series to determie covergece. The geeral priciple is this: if a positive series is bigger tha a positive diverget series, the it diverges, ad if a positive series is smaller tha a positive coverget series, the it coverges. For example, i the last sectio (Example ) we showed that coverges usig the Itegral Test. The we used the Itegral Test agai (Example ) to show that coverges. But, is smaller tha for all, so the covergece of is guarateed by the covergece of. While this approach should seem ituitively clear ad simple, we cautio the reader that it takes a lot of practice to become comfortable with comparisos. We state the formal test below. The Compariso Test. Suppose that 0 a b for sufficietly large. If a diverges, the b also diverges. If b coverges, the a also coverges. Before presetig the proof of the Compariso Test, ote the phrase sufficietly large. By 0 a b for sufficietly large, we mea that there is some umber N such that 0 a b for all N. This is just a formal way to say that we oly care about tails, ad should remid the reader of the Tail Observatio from Sectio.. Proof. Suppose that for all, 0 a b. This seems slightly weaker tha the result we have claimed, but the full result will the follow either by the Tail Observatio of Sectio. or by a easy adaptatio of this proof. Let s deote the th partial umber of a ad t deote the th partial sum of b, so s a a a, t b b b.

2 70 CHAPTER INFINITE SERIES From our hypotheses (that 0 a b for all ), we kow that s t for all. First suppose that a diverges. Because the terms a are oegative, the oly way that a ca diverge is if s as (why?). Therefore the larger partial sums t must also ted to as, so the series b diverges as well. Now suppose that b coverges, which implies by our defiitios that t b as. The sequece s is oegative ad mootoically icreasig because a 0 for all, ad 0 s t b so the sequece s has a limit by the Mootoe Covergece Theorem. This shows (agai, by the defiitio of series summatio) that the series a coverges. The Compariso Test leaves ope the questio of what to compare series with. I practice, however, this choice is usually obvious, ad we will almost always compare with a geometric series or a p-series. Our ext four examples demostrate the geeral techique. Example. Does the series verge? l coverge or di- Solutio. First ote that we probably should t try to apply the Itegral Test i this example the fuctio l x has a atiderivative, but it has bee proved that its atiderivative caot be expressed i terms of elemetary fuctios. However, the Compariso Test is easy to apply i this case. Note that 6 s a Sice l for, so l for. is a diverget p-series, diverges by compariso. l Example. Does the series l coverge or diverge? Solutio. This example ca be doe with the Itegral Test, but it s easier to use the Compariso Test. We kow that l for, so Sice l for. diverges, l must also diverge.

3 SECTION. THE COMPARISON TEST 7 Example. Does the series cos coverge or diverge? Solutio. We ca write this series as cos. The umerator of this fractio, cos, is oegative for all (this is importat sice we ca t apply the Compariso Test to series with egative terms) ad bouded by, so cos for. Therefore sice coverges (it is a coverget p-series), the smaller series must coverge as well. cos Example. Does the series coverge or diverge? l Solutio. For e 7.9, l, so for these values of, l. Sice is a coverget p-series, coverges by compariso. l Sometimes the iequalities we eed to apply the Compariso Test seem to go the wrog way. Cosider for example the series We would like to compare this series with the diverget series, but the terms i our series seem to be smaller tha the terms of aively apply the Compariso Test i this case.. Therefore we caot Example. Show that the series diverges.

4 7 CHAPTER INFINITE SERIES Solutio. We have that so the series diverges by compariso to , Our ext example displays a similar pheomeo. Note that, but we are still able to compare the series. Example 6. Show that the series coverges. Solutio. Because, we have that so the series coverges by compariso to. 8 9, Example 7. Show that the series 7 diverges. Solutio. We should expect this series to diverge, because the umerator is ad the deomiator behaves like, but the iequality goes the wrog way. By givig up a bit i the deomiator, however, we get the desired coclusio: 7 7 8, so the series we are iterested i diverges by compariso to the harmoic series. I Examples 7, we are really reidexig the series. This procedure is demostrated more formally i the example below ad i Exercises 8. Aother method for dealig with such problems, kow as the Limit Compariso Test, is discussed i Exercises 0. Example 8. Show that the series substitutio m. coverges by reidexig the series with the

5 SECTION. THE COMPARISON TEST 7 Solutio. We wat to compare this series to the series give by its leadig terms, (or some multiple of this), but the compariso seems to go the wrog way. By settig m, which is equivalet to m, we have m m m m m m m m 6m m. (Note here the chage i the lower boud, as i the previous example.) The iequality i the umerators (we wat to compare m m with m ) still goes the wrog way, but we ca take care of this by usig a slightly differet iequality: m m m m m 7m for m. The iequality i the deomiators does go the right way: m m 6m m m. Sice we have made the umerators larger ad the deomiators smaller, we have made the fractios larger, ad thus m m m m m 6m m m 7m 7 m m, m which implies by the Compariso Test that the series i questio coverges, because 7 m 7 m is a coverget p-series. Our ext example does t require reidexig, but does require a clever boud for l. So far we have used the facts that l for (i Example ) ad l for (i Example ). I fact, a much stroger upper boud holds. Let p be ay positive real umber. The by l Hôpital s Rule, we have lim x l x x p lim x x px p lim x px p 0. Recallig a fact from Sectio., this meas that lim l p 0 for every p 0. This i tur meas that for every p 0, l p for sufficietly large, a hady fact to have aroud for comparisos, as we demostrate ext. Example 9. Does the series l coverge or diverge?

6 7 CHAPTER INFINITE SERIES Solutio. As we showed above, l for sufficietly large (we could give a smaller boud, but is good eough here) ad, we ca use the compariso l. Because Test. is a coverget p-series, l coverges by the Compariso Our last example is cosiderably trickier tha the previous examples. The reader should pay attetio to the two themes it demostrates: first, whe dealig with a variable i a expoet, it is a good idea to use e ad atural log, ad secod, o matter how slowly a fuctio (such as l l ) goes to ifiity, it must evetually grow larger tha! Example 0. Does the series coverge or diverge? l l Solutio. ad l: We ow eed to test As the terms have a variable i the expoet, we first maipulate the usig e l l e l l l e l l l l l. for covergece. The approach from here o is similar to l l Example : for e e 68.8 (i.e., for large ), we have l l, so ad thus l l, l l coverges by compariso to the coverget p-series. If a series coverges by the Compariso Test, the we have the followig remaider estimate, which we coclude the sectio with. The Compariso Test Remaider Estimate. Let a ad b be series with positive terms such that a b for N. The for N, the error i the th partial sum of a, s, is bouded by b b : s a b b.

7 SECTION. THE COMPARISON TEST 7 Proof. By defiitio, s a a a. Now because the terms of a are positive, this is a a, ad sice we have assumed that N, a a b b, provig the estimate. Example. How may terms are eeded to approximate Solutio. We use the compariso to withi 0? for all to boud the error i approximatig be a good eough approximatio:. The first partial sum may ot s. The secod ad third partial sums are also ot guarateed to be as close to the true sum as required: s s but the fourth partial sum is withi 0:, 8, s 6 6. Therefore the aswer is that terms will certaily approximate the series withi 0.

8 76 CHAPTER INFINITE SERIES EXERCISES FOR SECTION. I Exercises, assume that a ad b are both series with positive terms.. If a b for sufficietly large ad b is coverget, what ca you say about a?. If a b for sufficietly large ad b is diverget, what ca you say about a?. If a b for sufficietly large ad b is coverget, what ca you say about a? e. If a b for sufficietly large ad b is diverget, what ca you say about a? Determie if the series i Exercises 8 coverge or diverge arcta Suppose that a is a coverget series with positive terms. Determie whether the series listed i Exercises 9 ecessarily coverge. If a series does t ecessarily coverge, give a example of a coverget series a with positive terms for which it diverges. It may be helpful to remember that there are oly fiitely may values of a at least, so these have o affect o the covergece of the series. a a a a si a a....! si I Exercises 8, use reidexig like we did i Examples 8 to determie if the give series coverge or diverge

9 SECTION. THE COMPARISON TEST Aother way to deal with problems like Exercises 8 is to apply the followig test. Usig the Compariso Test, determie if the series i Exercises 9 8 coverge or diverge l l l l l l l!! l 9. Costruct a example showig that the Compariso Test eed ot hold if a ad b are ot required to have positive terms. 0. If a,b 0 ad a ad b both coverge, show that the series a b coverges.. Prove that the series p si coverges for p. What about whe p? The Limit Compariso Test. Let ad b a be series with positive terms. If lim is a fiite b umber, the a ad b both coverge or both diverge. Exercise leads you through the proof of the Limit Compariso Test. After that, Exercises 8 preset applicatios, while Exercises 9 ad 0 exted the Limit Compariso Theorem to the case where the limit is 0 or. a. Suppose that lim c where c is a fiite b umber. Therefore there are positive umbers m ad M with m c M such that m a M for all large. Use this iequality ad the Compariso Test to derivate the Limit Compariso Test.. Show that the series from Example 8 coverges usig the Limit Compariso Test.. Does. Does 6. Does b coverge or diverge? coverge or diverge? 9 coverge or diverge? 7. Suppose that a 0 ad a 0. Show that si a coverges if ad oly if a coverges. 8. Suppose that 0 a for all. Prove that arcsi a coverges if ad oly if a coverges. 9. Let a ad b be series with positive terms. If a b 0 ad b coverges, prove that a coverges. 0. Let a ad b be series with positive terms. If a b ad b diverges, prove that a diverges.

10 78 CHAPTER INFINITE SERIES ANSWERS TO SELECTED EXERCISES, SECTION.. a coverges, by the Compariso Test. You caot coclude aythig. Coverges by compariso to 7. Coverges by compariso to 9. Coverges by compariso to. Coverges by compariso to π. Coverges by compariso to. Diverges by compariso to 7. Coverges by compariso to 9 9. Coverges by the Compariso Test: a a. Need ot coverge, cosider takig a. Sice a coverges, a for sufficietly large. For these values of, a a, so a coverges by the Compariso Test