The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

Transcription

1 460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that a series will coverge, ad oe has o guaratee the series ca either coverge or diverge. Which is which? Explai your reasoig. a. a a 0 b. c. d. a a a a a a The Ratio ad Root Tests Use the Ratio Test to determie whether a series coverges or diverges. Use the Root Test to determie whether a series coverges or diverges. Review the tests for covergece ad divergece of a ifiite series. The Ratio Test This sectio begis with a test for absolute covergece the Ratio Test. THEOREM 9.7 Ratio Test Let a be a series with ozero terms.. a coverges absolutely if. diverges if or a > a a. The Ratio Test is icoclusive if Proof To prove Property, assume that a a r < ad choose R such that 0 r < R <. By the defiitio of the it of a sequece, there exists some N > 0 such that a a < R for all > N. Therefore, you ca write the followig iequalities. a N < a N R a N < a N R < a N R a N < a N R < a N R < a N R <. a a a. a. The geometric series coverges, ad so, by the Direct Compariso Test, the series a N R a N R a N R... a N R... a N a N a N... a N... also coverges. This i tur implies that the series a coverges, because discardig a fiite umber of terms N ) does ot affect covergece. Cosequetly, by Theorem 9.6, the series a coverges absolutely. The proof of Property is similar ad is left as a exercise (see Exercise 98). NOTE The fact that the Ratio Test is icoclusive whe ca be see by comparig the two series ad. The first series diverges ad the secod oe coverges, but i both cases a. a a a

2 460_0906.qxd //04 :8 PM Page CHAPTER 9 Ifiite Series Although the Ratio Test is ot a cure for all ills related to tests for covergece, it is particularly useful for series that coverge rapidly. Series ivolvig factorials or expoetials are frequetly of this type. EXAMPLE Usig the Ratio Test Determie the covergece or divergece of 0!. STUDY TIP A step frequetly used i applicatios of the Ratio Test ivolves simplifyig quotiets of factorials. I Example, for istace, otice that!!!!. Solutio Because a you ca write the followig. a!, a!!!! 0 Therefore, the series coverges. EXAMPLE Usig the Ratio Test Determie whether each series coverges or diverges. a. b. 0! Solutio a. This series coverges because the it of a a is less tha. a a < b. This series diverges because the it of is greater tha. a a!! e > a a

3 460_0906.qxd //04 :8 PM Page 64 SECTION 9.6 The Ratio ad Root Tests 64 EXAMPLE A Failure of the Ratio Test NOTE The Ratio Test is also icoclusive for ay p-series. Determie the covergece or divergece of Solutio The it of is equal to. So, the Ratio Test is icoclusive. To determie whether the series coverges, you eed to try a differet test. I this case, you ca apply the Alteratig Series Test. To show that a a, let fx The the derivative is fx a a x x. x xx. a a Because the derivative is egative for x >, you kow that f is a decreasig fuctio. Also, by L Hôpital s Rule, x x x x x x x 0.. Therefore, by the Alteratig Series Test, the series coverges. The series i Example is coditioally coverget. This follows from the fact that the series a diverges by the Limit Compariso Test with a coverges., but the series TECHNOLOGY A computer or programmable calculator ca reiforce the coclusio that the series i Example coverges coditioally. By addig the first 00 terms of the series, you obtai a sum of about 0.. (The sum of the first 00 terms of the series a is about 7.)

4 460_0906.qxd //04 :8 PM Page CHAPTER 9 Ifiite Series The Root Test The ext test for covergece or divergece of series works especially well for series ivolvig th powers. The proof of this theorem is similar to that give for the Ratio Test, ad is left as a exercise (see Exercise 99). THEOREM 9.8 Root Test Let a be a series.. a coverges absolutely if. a diverges if or a > a <.. The Root Test is icoclusive if a. a. EXAMPLE 4 Usig the Root Test Determie the covergece or divergece of e. NOTE The Root Test is always icoclusive for ay p-series. Solutio You ca apply the Root Test as follows. a e e 0 < e Because this it is less tha, you ca coclude that the series coverges absolutely (ad therefore coverges). FOR FURTHER INFORMATION For more iformatio o the usefuless of the Root Test, see the article N! ad the Root Test by Charles C. Mumma II i The America Mathematical Mothly. To view this article, go to the website To see the usefuless of the Root Test for the series i Example 4, try applyig the Ratio Test to that series. Whe you do this, you obtai the followig. a a 0 e () e e e Note that this it is ot as easily evaluated as the it obtaied by the Root Test i Example 4.

5 460_0906.qxd //04 :8 PM Page 64 SECTION 9.6 The Ratio ad Root Tests 64 Strategies for Testig Series You have ow studied 0 tests for determiig the covergece or divergece of a ifiite series. (See the summary i the table o page 644.) Skill i choosig ad applyig the various tests will come oly with practice. Below is a set of guidelies for choosig a appropriate test. Guidelies for Testig a Series for Covergece or Divergece. Does the th term approach 0? If ot, the series diverges.. Is the series oe of the special types geometric, p-series, telescopig, or alteratig?. Ca the Itegral Test, the Root Test, or the Ratio Test be applied? 4. Ca the series be compared favorably to oe of the special types? I some istaces, more tha oe test is applicable. However, your objective should be to lear to choose the most efficiet test. EXAMPLE 5 Applyig the Strategies for Testig Series Determie the covergece or divergece of each series. a. b. c. 6 d. e. f. 4 g. e! 0 Solutio a. For this series, the it of the th term is ot 0 a as. So, by the th-term Test, the series diverges. b. This series is geometric. Moreover, because the ratio r 6 of the terms is less tha i absolute value, you ca coclude that the series coverges. c. Because the fuctio fx xe x is easily itegrated, you ca use the Itegral Test to coclude that the series coverges. d. The th term of this series ca be compared to the th term of the harmoic series. After usig the Limit Compariso Test, you ca coclude that the series diverges. e. This is a alteratig series whose th term approaches 0. Because a a, you ca use the Alteratig Series Test to coclude that the series coverges. f. The th term of this series ivolves a factorial, which idicates that the Ratio Test may work well. After applyig the Ratio Test, you ca coclude that the series diverges. g. The th term of this series ivolves a variable that is raised to the th power, which idicates that the Root Test may work well. After applyig the Root Test, you ca coclude that the series coverges.

6 460_0906.qxd //04 :8 PM Page CHAPTER 9 Ifiite Series Summary of Tests for Series Test Series Coditio(s) of Covergece Coditio(s) of Divergece Commet th-term a a 0 This test caot be used to show covergece. Geometric Series ar 0 r < r Sum: S a r Telescopig Series b b b L Sum: S b L p-series p p > p Alteratig Series a 0 < a a ad a 0 Remaider: R N a N Itegral ( f is cotiuous, positive, ad decreasig) a, a f 0 f x dx coverges f x dx diverges Remaider: 0 < R N < fx dx N Root a a < a > Test is icoclusive if a. Ratio a a a < a a > Test is icoclusive if a. a Direct Compariso a, b > 0 a 0 < a b ad b coverges 0 < b a ad b diverges Limit Compariso a, b > 0 a a L > 0 b ad b coverges a L > 0 b ad b diverges

7 460_0906.qxd //04 :8 PM Page 645 SECTION 9.6 The Ratio ad Root Tests 645 Exercises for Sectio 9.6 See for worked-out solutios to odd-umbered exercises. I Exercises 4, verify the formula !! k! k! kk 5... k k! k k! 5... k 5 k k!k k, k! k Numerical, Graphical, ad Aalytic Aalysis I Exercises ad, (a) verify that the series coverges. (b) Use a graphig utility to fid the idicated partial sum S ad complete the table. (c) Use a graphig utility to graph the first 0 terms of the sequece of partial sums. (d) Use the table to estimate the sum of the series. (e) Explai the relatioship betwee the magitudes of the terms of the series ad the rate at which the sequece of partial sums approaches the sum of the series I Exercises 5 0, 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) S S S 4 4!! 4! 4 5 4e (b) (d) (f) 0 S S S S I Exercises, use the Ratio Test to determie the covergece or divergece of the series..! ! !!. 4.! !! 7. 8.! 0 0! ! I Exercises 6, verify that the Ratio Test is icoclusive for the p-series !! p

8 460_0906.qxd //04 :8 PM Page CHAPTER 9 Ifiite Series I Exercises 7 50, use the Root Test to determie the covergece or divergece of the series l e ! l I Exercises 5 68, determie the covergece or divergece of the series usig ay appropriate test from this chapter. Idetify the test used cos l l! ! ! I Exercises 69 7, idetify the two series that are the same. 69. (a) 70. (a) 4 5 4! (b) 5 0! (b) (c) (c) 0 5! (a) 7. (a) 0! (b) (b)! (c) (c)! 0 I Exercises 7 ad 74, write a equivalet series with the idex of summatio begiig at ! I Exercises 75 ad 76, (a) determie the umber of terms required to approximate the sum of the series with a error less tha 0.000, ad (b) use a graphig utility to approximate the sum of the series with a error less tha k k k k! I Exercises 77 8, the terms of a series a are defied recursively. Determie the covergece or divergece of the series. Explai your reasoig. 77. a, a 78. a, a 5 4 a 79. a, a 80. a 5, a a 4, a a I Exercises 8 86, use the Ratio Test or the Root Test to determie the covergece or divergece of the series k a, a a l l 4 4 l 5 5 l a si a cos a k k

9 460_0906.qxd //04 :8 PM Page 647 SECTION 9.6 The Ratio ad Root Tests 647 I Exercises 87 9, fid the values of x for which the series coverges x x 0 4 x 0! 0 x 0 x x! Writig About Cocepts 9. State the Ratio Test. 94. State the Root Test. 95. You are told that the terms of a positive series appear to approach zero rapidly as approaches ifiity. I fact, a Give o other iformatio, does this imply that the series coverges? Support your coclusio with examples. 96. The graph shows the first 0 terms of the sequece of partial sums of the coverget series. Fid a series such that the terms of its sequece of partial sums are less tha the correspodig terms of the sequece i the figure, but such that the series diverges. Explai your reasoig. S Usig the Ratio Test, it is determied that a alteratig series coverges. Does the series coverge coditioally or absolutely? Explai. 99. Prove Theorem 9.8. (Hit for Property : If the it equals r <, choose a real umber R such that r < R <. By the defiitios of the it, there exists some N > 0 such that a < R for > N. 00. Show that the Root Test is icoclusive for the p-series p. 0. Show that the Ratio Test ad the Root Test are both icoclusive for the logarithmic p-series l p. 0. Determie the covergece or divergece of the series! x! whe (a) x, (b) x, (c) x, ad (d) x is a positive iteger. 0. Show that if a is absolutely coverget, the a a. 04. Writig Read the article A Differetiatio Test for Absolute Covergece by Yaser S. Abu-Mostafa i Mathematics Magazie. The write a paragraph that describes the test. Iclude examples of series that coverge ad examples of series that diverge. Putam Exam Challege 05. Is the followig series coverget or diverget? Show that if the series a a a... a... coverges, the the series a a a... a... coverges also.! 9 7! ! These problems were composed by the Committee o the Putam Prize Competitio. The Mathematical Associatio of America. All rights reserved. 98. Prove Property of Theorem 9.7.