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12-4Convergent andDivergent SeriesHISTORYonRThe Greek philosopher Zeno of Elea (c. 490–430 B.C.)proposed several perplexing riddles, or paradoxes. One of Zeno’sp li c a tiparadoxes involves a race on a 100-meter track between themythological Achilles and a tortoise. Zeno claims that even though Achilles can runtwice as fast as the tortoise, if the tortoise is given a 10-meter head start, Achilles willnever catch him. Suppose Achilles runs 10 meters per second and the tortoise aremarkable 5 meters per second. By the time Achilles has reached the 10-meter mark,the tortoise will be at 15 meters. By the time Achilles reaches the 15-meter mark, thetortoise will be at 17.5 meters, and so on. Thus, Achilles is always behind the tortoiseand never catches up.Ap Determinewhether a seriesis convergent ordivergent.l WorealdOBJECTIVEIs Zeno correct? Let us look at thedistance between Achilles and thetortoise after specified amounts oftime have passed. Notice that thedistance between the two contestantswill be zero as n approaches infinity10since lim n 0.n 2To disprove Zeno’s conclusion thatAchilles will never catch up to thetortoise, we must show that there is atime value for which this 0 differencecan be achieved. In other words, weneed to show that the infinite series1111 has a sum, or248Time(seconds)Distance Apart(meters)010110 52123210 2.541 12147410 1.2581 1 248. 8.110 0.625161111 10 2n2141815.limit. This problem will be solved inExample 5.Starting with a time of 1 second, the partial sums of the time series form the3 7 15sequence 1, , , , . As the number of terms used for the partial sums2 4 18increases, the value of the partial sums also increases. If this sequence ofpartial sums approaches a limit, the related infinite series is said to converge.If this sequence of partial sums does not have a limit, then the related infiniteseries is said to diverge.786Chapter 12Sequences and Series

Convergentand DivergentSeriesExampleThere are manyseries that beginwith the first fewterms shown inthis example. Inthis chapter,always assumethat the expressionfor the generalterm is thesimplest onepossible.If an infinite series has a sum, or limit, the series is convergent. If a seriesis not convergent, it is divergent.1 Determine whether each arithmetic or geometric series is convergent ordivergent.1111a. 248161This is a geometric series with r . Since r 1, the series has a limit.2Therefore, the series is convergent.b. 2 4 8 16 This is a geometric series with r 2. Since r Therefore, the series is divergent.c. 10 8.5 7 5.5 1, the series has no limit.This is an arithmetic series with d 1.5. Arithmetic series do not havelimits. Therefore, the series is divergent.When a series is neither arithmetic nor geometric, it is more difficult todetermine whether the series is convergent or divergent. Several differenttechniques can be used. One test for convergence is the ratio test. This test canonly be used when all terms of a series are positive. The test depends upon theratio of consecutive terms of a series, which must be expressed in general form.Let an and an 1 represent two consecutive terms of a series of positiveaRatio Testan 1n 1 exists and that r lim . The series isterms. Suppose lim n ann convergent if r 1 and divergent if rinformation.an1. If r 1, the test provides noThe ratio test is especially useful when the general form for the terms of aseries contains powers.Example2 Use the ratio test to determine whether each series is convergent ordivergent.a. 122438416First, find an and an 1.an n Then use the ratio test.n 1 n 12r lim n n 2n2nandn 1an 1 2n 1(continued on the next page)Lesson 12-4Convergent and Divergent Series787

n 1n 22nn r lim n 1 Multiply by the reciprocal of the divisor.n 1n1n 212n 22n 1r lim 12n 1nr lim lim n n Limit of a Product11 1nr lim 12 n Divide by the highest power of n andthen apply limit theorems.1 011212r or Since r 1, the series is convergent.b. 12233445nn 1n 1 n 2r lim nn n 1an n 1(n 1) 1n2 2n 1n 2nn n 1n 1n2 2n 1 n 2nn2 2nr lim 22n 1n 2an 1 or and11 2nnr lim 2n 1 Divide by the highest power of n andapply limit theorems.1 0 0r or 11 0Since r 1, the test provides no information.nThe ratio test is also useful when the general form of the terms of a seriescontains products of consecutive integers.Example3 Use the ratio test to determine whether the series1 is convergent or divergent.11 211 2 311 2 3 4First find the nth term and (n 1)th term. Then, use the ratio test.11 2 an r lim nand1 2 1 1 2 (n 1)n 1 1 2 n1 2 n (n 1)n 1 2 r lim 1n 1r lim or 0n Note that 1 2 (n 1) 1 2 n (n 1).Simplify and apply limit theorems.Since r 1, the series is convergent.788Chapter 121 (n 1)an 1 Sequences and Series

When the ratio test does not determine if a series is convergent or divergent,other methods must be used.Example is convergent or4 Determine whether the series 1 2345divergent.1111Suppose the terms are grouped as follows. Beginning after the second term,the number of terms in each successive group is doubled.111111111(1) 23456789161Notice that the first enclosed expression is greater than , and the second is21equal to . Beginning with the third expression, each sum of enclosed terms21is greater than . Since there are an unlimited number of such expressions,2the sum of the series is unlimited. Thus, the series is divergent.A series can be compared to other series that are known to be convergent ordivergent. The following list of series can be used for reference.Summary ofSeries forReference1. Convergent: a1 a1r a1r 2 a1r n 1 , r 12. Divergent: a1 a1r a1r 2 a1r n 1 , r 13. Divergent: a (a d ) (a 2d ) (a 3d ) 111114. Divergent: 1 2315. Convergent: 1 2p111 451 3p1 This series is known asnthe harmonic series.1 , p 1p nIf a series has all positive terms, the comparison test can be used todetermine whether the series is convergent or divergent.ComparisonTestExample A series of positive terms is convergent if, for n 1, each term of theseries is equal to or less than the value of the corresponding term ofsome convergent series of positive terms. A series of positive terms is divergent if, for n 1, each term of theseries is equal to or greater than the value of the corresponding term ofsome divergent series of positive terms.5 Use the comparison test to determine whether the following series areconvergent or divergent.4444a. 5791142n 3.The general term of this series is The general term of the divergentseries 1 is . Since for all n121314151n42n 31, 1 , thenseries is also divergent.454749411Lesson 12-4Convergent and Divergent Series789

b. 2 2 2 2 111315171(2n 1)11111convergent series 1 2 2 2 is 2 . Since 2234n(2n 1)1111all n, the series 2 2 2 2 is also convergent.1357The general term of the series is 2 . The general term of the1 forn2With a better understanding of convergent and divergent infinite series, weare now ready to tackle Zeno’s paradox.l WoreaAponldRExamplep li c a ti6 HISTORY Refer to the application at the beginning of the lesson. Todisprove Zeno’s conclusion that Achilles will never catch up to the tortoise,we must show that the infinite time series 1 0.5 0.25 has a limit.To show that the series 1 0.5 0.25 has a limit, we need to show thatthe series is convergent.1The general term of this series is n . Try using the ratio test for convergence2of a series.12an nan 1 1 and1 n 12r 1 2n1 22n 112n1 2n 112Since r 1, the series converges and therefore has a sum. Thus, there is a timevalue for which the distance between Achilles and the tortoise will be zero.You will determine how long it takes him to do so in Exercise 34.C HECKCommunicatingMathematicsFORU N D E R S TA N D I N GRead and study the lesson to answer each question.1. a. Write an example, of an infinite geometric series inwhich r 1.b. Determine the 25th, 50th, and 100th terms of yourSnseries.c. Identify the sum of the first 25, 50, and 100 termsof your series.d. Explain why this type of infinite geometric seriesdoes not converge.14121086422. Estimate the sum Sn of the series whose partial sumsare graphed at the right.790Chapter 12 Sequences and SeriesO1 2 3 4 5 6 7n

12232423. Consider the infinite series 3 4 .33233a. Sketch a graph of the first eight partial sums of this series.b. Make a conjecture based on the graph found in part a as to whether theseries is convergent or divergent.c. Determine a general term for this series.d. Write a convincing argument using the general term found in part c tosupport the conjecture you made in part b.4. MathJournal Make a list of the methods presented in this lesson and in theprevious lesson for determining convergence or divergence of an infinite series.Be sure to indicate any restrictions on a method’s use. Then number your list asto the order in which these methods should be applied.Guided PracticeUse the ratio test to determine whether each series is convergent or divergent.3125. 3 22223711156. 4812162347. Use the comparison test to determine whether the series is123convergent or divergent.Determine whether each series is convergent or divergent.15378. 41681611110. 3 1 22 223 21119. 2 2 2 122 22 3911. 4 3 412. EcologyAn underground storage container is leaking a toxic chemical. Oneyear after the leak began, the chemical had spread 1500 meters from its source.After two years, the chemical had spread 900 meters more, and by the end ofthe third year, it had reached an additional 540 meters.a. If this pattern continues, how far will the spill have spread from its sourceafter 10 years?b. Will the spill ever reach the grounds of a school located 4000 meters awayfrom the source? Explain.E XERCISESPracticeUse the ratio test to determine whether each series is convergent or divergent.AB444413. 93278148214. 10155481615. 2 2 2234222216. 2 33 44 53517. 1 1 2 31 2 3 4 5525318. 5 1 21 2 32 44 66 819. Use the ratio test to determine whether the series 2488 10 is convergent or divergent.16www.amc.glencoe.com/self check quizLesson 12-4 Convergent and Divergent Series791