4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then


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1 SECTION 2.6 THE RATIO TEST THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of course there are a great may series for which these two tests are ot ideally suited, for example, the series Itegratig the terms of this series would be difficult, especially sice the first step would be to fid a cotiuous fuctio which agrees with! (this ca be doe, but the solutio is ot easy). We could try a compariso, but agai, the solutio is ot particular obvious (ideed, those readers who solved Exercise 37 of the last sectio should feel proud). Istead, the simplest approach to such a series is the followig test due to Jea le Rod d Alembert (77 783). 4!. The Ratio Test. Suppose that a is a series with positive terms ad let a a. If the a coverges. If the a diverges. If or the it does ot exist the the Ratio Test is icoclusive. You shold thik of the Ratio Test as a geeralizatio of the Geometric Series Test. For example, if a ar is a geometric sequece the a a r, ad we kow these series coverge if ad oly if r. (Note that we will oly cosider positive series here; we deal with mixed series i the ext sectio.) I fact, the proof of the Ratio Test is little more tha a applicatio o the Compariso Test. Proof. If the the sequece a is icreasig (for sufficietly large ), ad therefore the series diverges by the Test for Divergece. Now suppose that. Choose a umber r sadwiched betwee ad : r. Because a a, there is some iteger N such that for all N. Set a a N. The we have 0 a a r a N ra N ar,
2 80 CHAPTER 2 INFINITE SERIES ad a N 2 ra N ar 2, ad i geeral, a N k ar k. Therefore for sufficietly large (amely, N), the terms of the series a are bouded by the terms of a coverget geometric series (sice 0 r ), ad so a coverges by the Compariso Test. Sice the Ratio Test ivolves a ratio, it is particularly effective whe series cotai factorials, as our first example does. Example. verge? Does the series 4! coverge or di s Solutio. First we compute : Sice a a 4! 4! 4 4!! 0, this series coverges by the Ratio Test a It is importat to ote that the Ratio Test is always icoclusive for series of the form polyomial polyomial. As a example, we cosider the harmoic series ad 2. Example 2. Show that the Ratio Test is icoclusive for ad 2. Solutio. For the harmoic series, we have. I order to evaluate this it, remember that we factor out the highest order term:, so the test is icoclusive. The series 2 fails similarly: ,
3 SECTION 2.6 THE RATIO TEST 8 ad agai we factor out the highest order term, leavig 2 2 so either series ca be hadled by the Ratio Test., 2 As Example 2 demostrates, kowig that a a is ot eough to coclude that the sequece coverges; we must kow that the it of this ratio is less tha. Example 3. Does the series coverge or diverge? Solutio. The ratio betwee cosecutive terms is a a as. Sice this it is less tha, we ca coclude that the series coverges by the Ratio Test. The last example could also be hadled by the Compariso Test, sice so the series coverges by compariso with a coverget geometric series. However, what if we moved the from the deomiator to the umerator: 0 4 2? Now the iequality i the compariso goes the wrog way, makig the Compariso Test much harder to use. O the other had, the it i the Ratio Test is uchaged (you should check this for yourself). I geeral, it is usually a good idea to try the Ratio Test o all series with expoetials (like 0 ) or factorials. Example 4. Does the series 2! 2! 0 6, coverge or diverge? 0 6 Solutio. Here the ratio betwee cosecutive terms is a a 2 2! 2! 2! 2!
4 82 CHAPTER 2 INFINITE SERIES as. Sice this it is greater tha (or ay other umber, for that matter), the series diverges by the Ratio Test. Our last example could be doe usig the Compariso Test (how?), but it is (probably) easier to use the Ratio Test. Example 5. Does the series or diverge? Solutio. terms is coverge I this case the ratio betwee cosecutive a a , so pullig out the highest order terms, we have a a as. Because this it is less tha, the series coverges by the Ratio Test. EXERCISES FOR SECTION 2.6 Exercises 4 give various values of a. a I each case, state what you coclude from the Ratio Test about the series a I Exercises 5 6, first compute a, a ad the use the Ratio Test to determie if the give series coverge or diverge
5 SECTION 2.6 THE RATIO TEST ! 2!! 2 2! 2!! 2!!! 99!!! 7. Fid a sequece a of positive (i particular, ozero) umbers such that both a ad a diverge. 8. Is there a sequece a satisfyig the coditios of the previous problem such that exists ad is ot equal to? a a A stroger test tha the Ratio Test, proved by Augusti ouis Cauchy ( ), is the followig. Use the Root Test to determie if the series i Exercises coverge or diverge l 2 2 Exercises 27 ad 28 show that the Root Test is a stroger test tha the Ratio Test. 27. Show that the Root Test ca hadle ay series that the Ratio Test ca hadle by provig that if a a exists the a. 28. Show that there are series that the Root Test ca hadle but that the Ratio Test caot hadle by cosiderig the series a where a 2 if is odd, 2 if is eve. The Root Test. Suppose that a 0 for all ad let a. The series a coverges if ad diverges if. (If the the Root Test is icoclusive.) I some cases where the ratio ad root tests are icoclusive, the followig test due to Joseph Raabe (80 859) ca prove useful. Our first task is to prove this result. 9. Copyig the begiig of the proof of the Ratio Test, give a proof of the Root Test. Raabe s Test. Suppose that a is a positive series. If there is some choice of p such that a a p
6 84 CHAPTER 2 INFINITE SERIES for all large, the a coverges. Exercises 29 3 ask you to prove Raabe s Test, while Exercises 32 ad 33 cosider a applicatio of the test. 29. Show that if p ad 0 x the px x p. This is called Beroulli s iequality, after Joha Beroulli ( ). Hit: Set f x px x p. Show that f 0 ad f x 0 for 0 x. Coclude from this that f x for all 0 x. 30. Assumig that the hypotheses of Raabe s Test hold ad usig Exercise 29, show that a a p b b where b p. 3. Rewrite the iequality derived i Exercise 30 as a b, b b use this to show that a Mb for some positive umber M ad all large, ad use this to prove Raabe s Test. 32. Show that the Ratio Test is icoclusive for the series 3 5 2k k Use Raabe s Test to prove that the series i Exercise 32 coverges.
7 SECTION 2.6 THE RATIO TEST 85 ANSWERS TO SEECTED EXERCISES, SECTION 2.6. The series diverges 3. The series coverges , so the series coverges by the Ratio Test. 7. 3, so the series diverges by the Ratio Test. a 9. 0 as, so the series coverges by the Ratio Test. a a. as, so the series coverges by the Ratio Test. a a 3. as, so the series diverges by the Ratio Test. a 5. The ratio here is a! a!. Recall from Example of Sectio 2. that the it of this ratio is Ratio Test because e. e, so the series diverges by the
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