· Created Date: 1/9/2001 9:45:59 AM

Chapter Nine

SERIES In this chapter we look at infinite sums,calledinfinite series. Westart in Section9.1with aparticular type of series,calleda geometricseries.Section9.2considersgeneralsequencesand seriesof constantsand what it meansforsuchseriesto converge.The teststhat allow usto determineconvergenceare inSections9.2and 9.3.Section??, intr oducespower series,in which the terms arepowersof� . Theseseriesconvergefor some � -valuesandnot for others; the radius of convergenceisintr oducedto identify the interval on which theseriesconverges.

426 Chapter Nine SERIES

9.1 GEOMETRIC SERIES

Thissectionintroducesinfinite seriesof constants,whicharesumsof theform�� The individual numbers,

�,�� or

�� etc.,arecalled terms in the series.To talkaboutthesum of theseries,wemustfirst explainhow to addinfinitely many numbers.

Let us look at the repeatedadministrationof a drug. In this example,the termsin the seriesrepresenteachdose;thesumof theseriesrepresentsthedruglevel in thebodyin thelongrun.

Repeated Drug DosageA personwith anearinfectionis told to take antibiotictabletsregularly for severaldays.Sincethedrug is beingexcretedby the bodybetweendoses,how canwe calculatethe quantityof the drugremainingin thebodyatany particulartime?

To bespecific,let’ssupposethedrugis ampicillin (acommonantibiotic)takenin 250mgdosesfour timesa day(that is, every six hours).It is known thatat theendof six hours,about

� �of the

drug is still in thebody. Whatquantityof thedrug is in thebody right after the tenthtablet?Thefortieth?

Let !#" representthequantity, in milligrams,of ampicillin in thebloodright afterthe $&%(' tablet.Then! �*),+�- ),+�- mg! � ) +�- �./ �� 102 3�4 5

Remnantsof first tablet

� +�- 263�465New tablet

),+�7 mg! � ) ! � .( �� 108� +�- ) . +�- �./ �� 90:� +�- �0&.( �� 10;� +�- ) +�- �.( �� 90 � � +�- �.( �� 902 3�4 5Remnantsof first andsecondtablets

� +�- 263�465New tablet

),+�7 �� mg! ) ! � .( �� 108� +�- )=<>+�- �.( �� 90 � � +�- �.( �� 90:� +�- �?@./ �� 90&� +�- ),+�- �.( �� 90 � � +�- �.( �� 90 � � +�- �.( �� 102 3A4 5Remnantsof first, second,andthird tablets

� +�- 2B3A465New tablet

),+�7 �� 7 mg

Lookingat thepatternthatis emerging,weguessthat!#C )D+�- �.( �� 10 � +�- �.( �� 10 � � +�- �.( �� 10 � � +�- �.( �� 10;� +�- ! �FE )D+�- �.( �� 10FGH� +�- �.( �� 10JIK��6� +�- �.( �� 90:� +�- ��Noticethatthereare10 termsin thissum—onefor everytablet—but thatthehighestpowerof 0.04is theninth,becauseno tablethasbeenin thebodyfor morethan9 six-hourtimeperiods.(Do youseewhy?)Now supposeweactuallywantto find thenumericalvalueof ! �FE . It seemsthatwehaveto add10 terms—andif wewantthevalueof ! E , wewouldbefacedwith adding40 terms:! EL),+�- �.( �� 90 � G � +�- �./ �� 10 � I ��6� +�- �.( �� 90:� +�- ��

Fortunately, there’sa betterway. Let’s startwith ! �FE .! �ME )D+�- �./ �� 10 G � +�- �./ �� 10 I � +�- �./ �� 10JNH��6� +�- �.( �� 10 � � +�- �./ �� 10&� +�- ��Noticetheremarkablefactthatif yousubtract

.( �� 90 ! �FE from ! �FE , agreatmany terms(all but two,in fact)dropout.Firstmultiplying by

�� , weget.( �� 10 ! �ME�)D+�- �.( �� 10 �ME � +�- �./ �� 10 G � +�- �.( �� 90 I ��O� +�- �.( �� 10 � � +�- �.( �� 90 � � +�- �.( �� 10��

9.1 GEOMETRIC SERIES 427

Subtractinggives ! �ME*P .( �� 10 ! �MEL),+�- PQ+�- �./ �� 10 �ME �Factoring ! �ME ontheleft andsolvingfor ! �FE gives! �FE .J� P �� 10 )D+�- < � P .( �� 90 �FE ?! �MEL) +�- < � P .( �� 90 �FE ?� P �� This is called the closed-form expressionfor ! �FE . It is easyto evaluateon a calculator, giving! �FE ),+�7 �� + (to two decimalplaces).Similarly, ! E is givenin closed-formby! E ) +�- < � P .( �� 10 E ?� P �� Evaluatingthis on a calculatorshows ! E�)R+�7 �� + , which is thesame(to two decimalplaces)as! �FE . Thusaftertentablets,thevalueof !#" appearsto havestabilizedat justover260mg.

Lookingat theclosed-formsfor ! �ME and ! E , wecanseethat,in general,! " mustbegivenby! " ) +�- �.S� P .( �� 10 " 0� P �� What Happens as TVUXW ?

Whatdoesthis closed-formfor ! " predictaboutthe long-runlevel of ampicillin in thebody?As$DY[Z , the quantity./ �� 90 " Y

. In the long run, assumingthat 250 mg continueto be takeneverysix hours,thelevel right aftera tabletis takenis givenby! " ) +�- �.S� P .( �� 90 " 0� P �� Y +�- �.F� P 10� P �� ]\ +�7 �� + �

The Geometric Series in GeneralIn thepreviousexamplewe encounteredsumsof theform ^ � ^1_ � ^1_ � �,��B� ^1_ I � ^1_ G (with^ )�+�- and _ ) ��

). Sucha sumis calleda finite geometric series. A geometricseriesis onein which eachterm is a constantmultiple of the onebefore.The first term is ^ , andthe constantmultiplier, or common ratio of successive terms,is _ . (In ourexample, )D+�- and _ ) ��

.)

A finite geometricserieshastheform^ � ^1_ � ^1_ � ��B� ^1_ "9` � � ^1_ "9` � �An infinite geometricserieshastheform^ � ^1_ � ^1_ � ��6� ^�_ "9` � � ^1_ " ` � � ^1_ " ��

The“��

” at theendof thesecondseriestellsusthattheseriesis goingon forever—in otherwords,thatit is infinite.

Sum of a Finite Geometric SeriesThesameprocedurethatenabledusto find theclosed-formfor ! �FE canbeusedto find thesumofany finite geometricseries.Supposewe write a;" for thesumof thefirst $ terms,which meansupto thetermcontaining_ "9` � :a " ) ^ � ^1_ � ^1_ � ��6� ^1_ "9` � � ^1_ "9` � �


Multiply a8" by _ : _:a8" ) ^1_ � ^1_ � � ^1_ � ��6� ^1_ " ` � � ^1_ " �Now subtract_:a;" from a;" , whichcancelsoutall termsexceptfor two, givinga " P _&a " ) ^ P ^1_ ".F� P _ 0 a " ) ^ .J� P _ " 0A�Provided _cb) �

, wecansolve to find a closedform for a " asfollows:

Thesumof a finite geometricseriesis givenbya " ) ^ � ^1_ � ^1_ � �d��A� ^1_ "9` � ) ^ .F� P _ " 0� P _ � provided _cb) �.

Notethatthevalueof $ in theformulafor a8" is thenumberof termsin thesum a;" .

Sum of an Infinite Geometric SeriesIn theampicillin example,we foundthesum ! " andthenlet $eYfZ �

We do thesamehere.Thesum ! " , which shows theeffect of thefirst $ doses,is anexampleof a partial sum. Thefirst threepartialsumsof theseries

� ^1_ � ^1_ � ��6� ^�_ "9` � � ^1_ " ��area � ) ^a � ) ^ � ^1_a � ) ^ � ^1_ � ^1_ � �

To find thesum, a , of this infinite series,we considerthepartialsum, a " , of thefirst $ terms.Theformulafor thesumof afinite geometricseriesgivesa " ) ^ � ^1_ � ^�_ � ��B� ^1_ " ` � ) ^ .J� P _ " 0� P _ �Whathappensto a " as $gYhZ ? It dependson thevalueof _ . If i _ji�k �

, then _ " Y as $lYhZ ,

so monop"�qsr a;" ) montp"�qsr ^ .F� P _ " 0� P _ ) ^ .F� P �0� P _ ) ^� P _ �Thus,provided i _jiLk �

, as $uYvZ the partial sums a;" approacha limit of ^�w .J� P _ 0 . Whenthis happens,we definethesumof the infinite geometricseriesto be that limit andsaytheseriesconverges to ^�w .J� P _ 0 .

For i _ji k �, thesumof the infinite geometricseriesis givenbya ) ^ � ^�_ � ^1_ � ��B� ^1_ "9` � � ^�_ " �� ) ^� P _ �

If, on theotherhand, i _ji�x �, then _ " andthepartialsumshaveno limit as $yYzZ (if ^�b)

).In thiscase,wesaytheseriesdiverges. If _�x �

, thetermsin theseriesbecomelargerandlargerinmagnitude,andthepartialsumsdivergeto

� Z (if ^gx ) or P Z (if ^{k

). When _�k P � , thetermsbecomelargerin magnitude,thepartialsumsoscillateas $|Y}Z , andtheseriesdiverges.


Whathappenswhen _ ) �?Theseriesis

^ � ^ � ^ � ^ ��

andif ^�b) , thepartialsumsgrow withoutbound,andtheseriesdoesnotconverge.When _ )RP � ,

theseriesis

^ P ^ � ^ P ^ � ^ P ��

and,if ^�b) , thepartialsumsoscillatebetween and0, andtheseriesdoesnotconverge.


Example 1 For eachof thefollowing infinitegeometricseries,find severalpartialsumsandthesum(if it exists).

(a)�~� �+ � ��

(b)�~� + �Q�L��

(c) 7�Pc+ � +� P +� � ++�� P ��Solution (a) Thisseriesmaybewritten �~� �+ �=� �+&� � �� +;� � ��

which we canidentify asa geometricserieswith ^ ) �and _ ) �� , so a ) �� P .F� w + 0 )�+ .

Let’s checkthisby finding thepartialsums:a � ) �a � ) �~� �+ ) � + )�+�P �+a � ) �~� �+ � �� ) �� ),+�P ��a ) �~� �+ � �� ) � -� )D+�P ��a C ) �~� �+ � �� 7 ) �� 7 )�+�P �� 7 �Theformulafor a " gives a " ) � P .6�� 0 "� P �� )�+�P � �+8� "9` � �Thus,thepartialsumsarecreepingup to thevalue a )�+ , so a " Y + as $yYzZ .

(b) Thepartialsumsof this geometricseries(with ^ ) �and _ )�+ ) grow without bound,sothe

serieshasnosum: a �*) �a � ) �~� +s) �a � ) �~� + �Q� )D�a ) �~� + �Q�L�� ) � -a C ) �~� + �Q�L�� 7s) ��Theformulafor a " gives a;" ) � Pc+ "� PQ+ )D+ " P ��

(c) This is aninfinite geometricserieswith ^ )�7 and _ )RP �� . Thepartialsums,a � )�7 � � � a � ) �� a � \ �� 71� � a \ �� a C \ �� -�+ � a;� \ �� appearto beconvergingto

�� - . This turnsout to becorrectbecausethesumisa ) 7� P . P � w �10 ) �� - �


Regular Deposits into a Savings AccountPeoplewho save money oftendo soby puttingsomefixedamountasideregularly. To bespecific,suppose� �6 � � is depositedeveryyearin a savingsaccountearning- � a year, compoundedannu-ally. Whatis thebalance,��" , in dollars,in thesavingsaccountright afterthe $&%(' deposit?

As before,let’s startby lookingat thefirst few years:� � ) �6 � � � � ) � � .J�� - 08�� ) �� .J�� - 02 3A4 5Originaldeposit

� �6 � � 2 3A4 5New deposit� � ) � � .J�� - 08�� ) �6 � � �.F�� - 0 � �� .J�� - 02 3�4 5

First two deposits

� �6 � � 2 3A4 5New deposit� ) � � .J�� - 08�� ) �6 � � �.F�� - 0 � �� .J�� - 0 � ��6 � � �.F�� - 02 3�4 5

First threedeposits

� �� 2 3A4 5New deposit

Observingthepattern,wesee� " ) �� .F�� - 0 "9` � �� .F�� - 0 "9` � ��6�� .J�� - 0&�� So � " is afinite geometricserieswith ^ ) ��

and _ ) �� - . Thuswehave��" ) �� .S� P .F�� - 0 " 0� P �� - �We canrewrite thissothatboththenumeratoranddenominatorof thefractionarepositive:� " ) �� .M.J�� - 0 " P �B0�� -LP � �What Happens as TVUXW ?

Commonsensetells you that if you keepdepositing � �� in an accountand it keepsearninginterest,yourbalancegrowswithoutbound.This is whattheformulafor � " showsalso:

.F�� - 0 " YZ as $yYzZ , so � " hasno limit. (Alternatively, observethattheinfinite geometricseriesof which� " is a partialsumhas_ ) �� - , which is greaterthan1, sotheseriesdoesnotconverge.)

Exercises and Problems for Section 9.1Exercises

In Problems1–10,decidewhichof thefollowing aregeomet-ric series.For thosewhichare,give thefirst termandtheratiobetweensuccessive terms.For thosewhich are not, explainwhy not.

1. �@� ��#� �� Q�� 2. � � ��#� ��s� �� s�c��3. � �l�A� � � �� *� ��4. � � � � ��#� �� c��5.

� �� 9��c�� 6. ��

7.�¡� �:�*� � � � � �*� � � �� 8.

�@� � � �|� � � ��¢£�c��A�9.

� � �6¤ � � ¤ � �Q¥ ¤ � �� ¤ � �c�� 10.� � ��¤ ��¦ ��¤¡§ � �¨¦ ��¤¡§ � ��A�

11. Find the sumof the se-riesin Problem6.




Find thesumof theseriesin Problems15–18.

15.� � � �£� �� A� � �� 16.

� � � �&� �� &�L�� ©(ª


17. «¬B® � � ��

18. «¬ 6® ª � � �� Problems

19. Thisproblemshowsanotherwayof deriving thelong-runampicillin level. Supposethat in the long run the ampi-cillin levels off to ¯ mg right aftereachtabletis taken.Six hourslater, right beforethe next dose,therewill belessampicillin in thebody. However, if stabilityhasbeenreached,theamountof ampicillin thathasbeenexcretedmustbeexactly 250mg becausetakingonemoretabletmustraisethe level backto ¯ mg. Usethis to solve for¯ .

20. Figure9.1shows thequantityof thedrugatenololin theblood as a function of time, with the first doseat time°�± �

. Atenolol is taken in 50 mg dosesoncea day tolowerbloodpressure.

� � � � �¯ ª

°(time, days)

² (quantity, mg) ¯ © ¯ � ¯ � ¯ �³ © ³ � ³ � ³ �

Figure 9.1

(a) If thehalf-life of atenololin theblood is 6.3 hours,what percentageof the atenololpresentat the startof a24-hourperiodis still thereat theend?

(b) Findexpressionsfor thequantities ª , ¯ © , ¯ � , ¯ � ,´�´�´ , and ¯ shown in Figure9.1.Write theexpres-sionfor ¯ in closed-form.

(c) Find expressionsfor the quantities³ © , ³ � , ³ � , ´�´�´ ,

and³

shown in Figure9.1. Write the expressionfor

³ in closed-form.

21. On page426,you saw how to computethequantity ¯ mg of ampicillin in thebodyright after the µ�¶¸· tabletof250mg,takenonceevery six hours.

(a) Do a similar calculation for³

, the quantity ofampicillin (in mg) in the body right before the µ ¶¸·tabletis taken.

(b) Express³

in closedform.(c) What is ¹»º»¼B½ « ³

? Is this limit the same as¹»º»¼B½ « ¯ ? Explain in practicaltermswhy your an-

swermakessense.

22. Draw agraphlike thatin Figure9.1for 250mgof ampi-cillin takenevery 6 hours,startingat time

°¾± �. Puton

thegraphthevaluesof ¯ © , ¯ � , ¯ �6¿ ´�´�´ introducedin thetext onpage426andthevaluesof

³ © , ³ � , ³ � ¿ ´�´�´ calcu-latedin Problem21.

23. A ball is droppedfrom a heightof 10 feetandbounces.Eachbounceis �� of the height of the bouncebefore.Thus after the ball hits the floor for the first time, theball risesto a heightof

�A� ¦�� § ±=À ´ � feet, andafter ithits the floor for the secondtime, it risesto a heightofÀ ´ � ¦�� § ± �A� ¦�� § � ± � ´ � �6� feet.

(a) Find an expressionfor the heightto which the ballrisesafterit hits thefloor for the µ ¶¸· time.

(b) Find anexpressionfor thetotal verticaldistancetheball hastraveledwhenit hits the floor for the first,second,third, andfourth times.

(c) Find anexpressionfor thetotal verticaldistancetheball hastraveledwhen it hits the floor for the µ ¶¸·time.Expressyouranswerin closed-form.

24. Youmightthink thattheball in Problem23keepsbounc-ing foreversinceit takesinfinitely many bounces.This isnot true!

(a) Show that a ball droppedfrom a height of Á feetreachesthegroundin

©�1Â Á seconds.(b) Show thattheball in Problem23stopsbouncingaf-

ter�� Â �A� � �� Â ��9Ã ��{Ä ��@�cÅ �BÆ �&ÇÉÈ �6�seconds

25. Supposethat ¥ � c/ of every dollarspentin theUS is spentagain in the US. If the Federalgovernmentpumpsanextra Ê � billion into theeconomy, how muchadditionalspendingoccursasaresult?

26. This problemillustrateshow bankscreatecreditandcantherebylend out moremoney thanhasbeendeposited.Supposethatinitially Ê ��6� is depositedin a bank.Expe-riencehasshown bankersthatonaverageonly

�BËof the

money depositedis withdrawn by theownerat any time.Consequently, bankersfeel freeto lendout ¥ �6Ë of theirdeposits.Thus Ê�¥ � of theoriginal Ê �A�� is loanedout toothercustomers(to starta business,for example).ThisÊ�¥ � will becomesomeoneelse’s incomeand,soonerorlater, will beredepositedin thebank.Then ¥ �6Ë of Ê�¥ � ,or Ê�¥ � ¦ � ´ ¥ ��§ ± Ê � � ´ � � , is loanedout againandeventu-ally redeposited.Of the Ê � � ´ � � , thebankagainloansout¥ ��Ë , andsoon.

(a) Find the total amountof money depositedin thebankasa resultof thesetransactions.

(b) Thetotalamountof money depositeddividedby theoriginaldepositis calledthecredit multiplier. Calcu-latethecreditmultiplier for thisexampleandexplainwhatthisnumbertellsus.

9.2 CONVERGENCE OF SEQUENCES AND SERIES 433

27. This problemdealswith the questionof estimatingthecumulative effect of a tax cut on a country’s economy.Supposethegovernmentproposesataxcut totaling Ê �A�6�million. We assumethat all the peoplewho have ex-tra money to spendwould spend80

Ëof it and save

20Ë

. Thus,of theextra incomegeneratedby thetaxcut,Ê ��6� ¦ � ´ �B§ million± Ê � � million would be spentand

so becomeextra incometo someoneelse.Assumethat

thesepeoplealsospend� � Ë

of their additionalincome,or Ê � � ¦ � ´ �B§ million, andsoon. Calculatethetotal addi-tionalspendingcreatedby sucha taxcut.

28. Supposethegovernmentproposesa taxcutof Ê ��6� mil-lion as in Problem27, but that economistsnow predictthatpeoplewill spend90

Ëof theirextraincomeandsave

only 10Ë

. How muchadditionalspendingwouldbegen-eratedby thetaxcutundertheseassumptions?

9.2 CONVERGENCE OF SEQUENCES AND SERIES

We now considergeneralseriesin which eachterm ^ " is a number. The seriescan be writtencompactlyusinga

¬signasfollowsr¬"�Ì � ^1" ) ^ � � ^ � � ^ � ��6� ^9" ��

For any particularvaluesof ^ and _ , thegeometricseriesis sucha series,with generalterm ^ " )^1_ "9` � . As in Section9.1,we investigateconvergenceof a seriesusingthepartialsum, a " , of thefirst $ terms,which is definedasa;" ) "¬ Í Ì � ^ Í ) ^ � � ^ � ��B� ^9" �

Convergence of SequencesThepartialsumsformasequence, orstringof numbers.1 A sequencecanbewritten a � � a � � a � � �� a " � �� ,where a " is calledthe general term. The limit of the sequenceis definedin a way similar to thelimit of a function;seealsoProblem20.

Thesequencea � � a � � a � � �� a " � �� hasa limit Î , written

mtnop"�qsr a " ) Î , if wecanmake a "ascloseto Î aswepleasefor all sufficiently large $ .

To calculatethe limit of a sequence,we usewhat we know aboutthe limits of functions,in-cludingthepropertiesin Theorem2.1andthefollowing facts:Ï montp"�q�r _ " )

if i _ji�k �Ï montp"�q�r � wB$ ) Example 1 Find thelimits of thefollowing sequences,if they exist.

(a) a " ) ./ �� 0 " (b) a " ) � P{Ð `:"�~� Ð `:" (c) a " ) �~��. P �60 "1In everydayEnglish,thewords“sequence”and“series”areusedinterchangeably. In mathematics,they have different

meaningsandcannotbeinterchanged.


Solution (a) Since./ �� 0 " decreasesto 0 as $ increases,wehave

mtnop"�qsr ./ �� 0 " ) .

(b) SinceÐ `&" Y as $|YÑZ , wehave

mtnop"�qsr a " ) �.

(c) Since. P �60 " ) �

if $ is evenand. P �B0 " )RP � if $ is odd,a �*) � a � )D+ � a � ) � a )�+ � �� andsoon

�Thesequencea " doesnothavea limit.

To show thata sequencehasa limit, thefollowing resultis oftenuseful.

Theorem 9.1: Convergenceof an Increasing,BoundedSequence

If a sequencea " , for $ ) � � + � � � �� , is increasingandboundedabove,then

monop"�qsr a " exists.

To understandthis theoremgeometrically, seeFigure9.2. Sayingthat a " is boundedabovemeansthatthevaluesof a " arelessthansomenumberÒ ; thatis, a "|Ó Ò . Since a " is increasingandboundedabove, thevaluesof a " must“pile up” at somenumberlessthanor equalto Ò . Thisnumberis thelimit.2 Ô

�Ô�

Ô�Ô Õ

Ô ©Figure 9.2: Valuesof Ö for µ ± � ¿ � ¿ �� ¿ �A�

Convergence of SeriesTo investigatetheconvergenceof a series,weconsiderthesequenceof partialsumsa � � a � � a � � �� a " � ��If a;" hasa limit as $yYzZ , thenwedefinethesumof theseriesto bethatlimit.

If

montp"�q�r a " exists,say

mtnop"�qsr a " ) a , thenwe saythe seriesr¬"�Ì � ^ " convergesandthat its

sumis a . We write

r¬"�Ì � ^9" ) a . If

monop"�qsr a;" doesnotexist, wesaythattheseriesdiverges.

Visualizing Series

We canvisualizethetermsof a seriesasin Figure9.3. In this figure,we assume "g× for all $ ,

soeachrectanglehasarea " . Thentheseriesconvergesif the total areaof therectanglesis finiteandthesumof theseriesis the total areaof therectangles.This is similar to an improperintegralØ rEÚÙ . _ 0 Û _ , in which theareaunderthegraphof Ù canbefinite, evenonaninfinite interval.

2Seetheonlinesupplementfor aproof.


Ô Ô Ô�Ô�Ô �Ô ©

. . . . . .� �É� � � µFigure 9.3: Heightandareaof the µ ¶¸· rectangleis

Ô Herearesomepropertiesthatareusefulin determiningwhetheror nota seriesconverges.

Theorem 9.2: ConvergencePropertiesof Series

1. Ifr¬"�Ì � ^ " and

r¬"�Ì ��Ü " convergeandif Ý is a constant,thenÏ r¬"�Ì � . ^ " � Ü " 0 convergesto

r¬"�Ì � ^ " � r¬"�Ì ��Ü " .Ï r¬"�Ì � Ý ^ " convergesto Ý r¬"�Ì � ^ " .

2. Changingafinite numberof termsin aseriesdoesnotchangewhetherornotit converges,althoughit maychangethevalueof its sumif it doesconverge.

3. Ifr¬"�Ì � ^1" converges,then

mtnop"�qsr ^9" ) ��4. If

r¬"�Ì � ^ " diverges,then

r¬"�Ì � Ý ^ " divergesif Ýyb) .

For proofsof theseproperties,seeProblems21–24.As for improperintegrals,theconvergenceof aseriesis determinedby its behavior for large $ . (Seethe“behaveslike” principleonpage374.)From Property2 we seethat, if Ý is a positive integer, then Þ r"�Ì � ^ " and Þ r"�Ì£ß ^ " eitherbothconvergeor bothdiverge.Thus,if all we careaboutis theconvergenceof a series,we canomit thelimits andwrite Þ ^9" .

Property3 often is usedto tell us thata seriesdoesnot converge.If the termsdo not go to 0,theserieshasnochanceof converging.Warning: Knowing that

monop"�qsr ^9" ) is not enoughto ensureconvergence.(SeeExample3.)

Example 2 Doestheseries¬ .F� PcÐ `:" 0 converge?

Solution Sincethe termsin the series, " ) � PàÐ `&" tendto 1, not 0, as $DY[Z , the seriesdivergesbyPropery3 of Theorem9.2.


Comparison of Series and IntegralsWe investigatetheconvergenceof someseriesby comparisonwith animproperintegral.Thehar-monic series is theinfinite series�~� �+ � �� ¡� �$ ��Convergenceof thissumwouldmeanthatthesequenceof partialsumsa �*) � � a � ) �~� �+ � a � ) �~� �+ � �� a;" ) �~� �+ � �� ,��6� �$ � ��tendsto a limit as $yYzZ . Let’s look at somevalues:a �*) � � a �FE \ + � �� a �FEáE \ - �o�� a �FE�EáE \ � � �� a �ME�EáE�E \ �� Thegrowth of thesepartialsumsis slow, but they do in factgrow without bound,so theharmonicseriesdiverges.This is justifiedin thefollowing exampleandin Problem25.

Example 3 Show thattheharmonicseries�~�� w + �� w �*�� w ��

diverges.

Solution The ideais to approximateØ r� .F� wB_ 0�Û _ by a left-handsum,wherethe terms

� � � w + � � w � � �� areheightsof rectanglesof base1. In Figure9.4,thesumof theareasof the

�rectanglesis largerthan

theareaunderthecurvebetween_ ) �and _ ) �

, andthesamekind of relationshipholdsfor thefirst $ rectanges.Thus,wehavea " ) �~� �+ � �� 6� �$ x�â "�ã �� _ Û _ ) moä . $ ��60Since

moä . $ �,�60getsarbitrarily largeas $lYÑZ , sodo thepartialsums,a " . Thus,thepartialsums

haveno limit, sotheseriesdiverges.

� � � �å � ± � Æ �æ Area

± �æ Area± � Æ6�æ Area

± � Æ�� Rectangles showing� � ©� � ©�sç Ø �© ©è¾é � ± ¹»ê ��

Figure 9.4: ComparingtheharmonicseriestoØ «© ¦ � Æ � § é �

Noticethattheharmonicseriesdiverges,eventhough

montp"�qsr ^ " ) mtnop"�qsr �$ ) .

Example 4 By comparisonwith theimproperintegralØ r� .J� wB_ � 0 Û _ , show thatthefollowing seriesconverges:r¬"�Ì � �$ � ) �~� ��


� � � � �� Æ �� Æ ¥� Æ �� ± � Æ � �

æ Area± � Æ �æ Area

± � Æ ¥æ Area± � Æ �A�

Area± � Æ6�6�æ

Shaded rectangles show�� ¥ � ��A� � ��6�ìë â «ª �� é ��

�

Figure 9.5: ComparingÞ «B® © � Æ µ�� toØ «© ¦ � Æ � � § é �

Solution Sincewewantto show that

r¬"�Ì � � wB$ � converges,wewantto show thatthepartialsumsof thisseries

tendto a limit. We do this by showing that thesequenceof partialsumsincreasesandis boundedabove,soTheorem9.1applies.

Eachsuccessive partialsumis obtainedfrom thepreviousoneby addingonemoretermin theseries.Sinceall thetermsarepositive,thesequenceof partialsumsis increasing.

To show thatthepartialsumsofr¬"�Ì � � wB$ � arebounded,we considertheright-handsumrepre-

sentedby theareaof therectanglesin Figure9.5.We startat _ ) �, sincetheareaunderthecurve

is infinite for Ó _ Ó �

. Theshadedrectanglesin Figure9.5suggestthat:�� 7 ��B� �$ � Ó â r� �_ � Û _ �Theareaunderthegraphis finite, sinceâ r� �_ � Û _ ) monopí qsr â í� �_ � Û _ ) monopí qsr � P �Ü �� ) ��To get a " , weadd1 to bothside,givinga;" ) �~� �� 7 ��6� �$ � Ó �~� â r� �_ � Û _ )D+ �Thus,thesequenceof partialsumsis boundedaboveby 2. Hence,by Theorem9.1thesequenceofpartialsumsconverges,sotheseriesconverges.

Noticethatwe have shown that theseriesin thepreviousexampleconverges,but we have notfoundits sum.Theintegralgivesusa boundon thepartialsums,but it doesnot giveusthelimit ofthepartialsums.Eulerprovedtheremarkablefactthatthesumis î � w 7 .

Themethodof Examples3 and4 canbeusedto provethefollowing theorem.SeeProblem28.

Theorem 9.3: The Integral Test

Supposeï × and Ù . _ 0 is a decreasingpositive function, definedfor all _ ×

, with^1" ) Ù . $ 0 for all $ .Ï If â rð Ù . _ 0 Û _ converges,then Þ ^ " converges.Ï If â rð Ù . _ 0 Û _ diverges,then Þ ^ " diverges.


Theintegral testcanbeusedto show that Þ r"�Ì � � wB$:ñ convergesfor ò�x �. SeeProblem13.


Problems1–8give expressionsfor Ö , the generaltermof asequence.Find thelimit of eachsequence,if it exists.

1. ¦ � ´ ��§ 2.

� 3. ¦ �¾� ´ �6§ 4.

� �yó¡ô � 5. õ�ö�÷�¦¸ø�µ §6.

� � 7.

� � � µ� � À µ8.

� µ#�Q¦ �*� § �� µ � ¦ �*� § �

Do theseriesin Problems9–12convergeor diverge?

9. «¬ 6® © µµ#� �10. «¬ 6® ª ó ô 11. «¬ 6® © ó 12. «¬ 6® © µ8¦¸µ#� � §Â µ � � � µ �

Problems

13. Usetheintegral testto show that «¬B® © � Æ µ9ù(a) Convergesif ú ç �(b) Divergesif úüû �

.

14. Show that «¬B® © ¹»êHµµ is divergent.

By identifyingeachoneasaleft- or right-handapproximationof an improperintegral, decidewhetherthe seriesin Prob-lems15–19converge.

15.� � �� ¥ � ��A� � ��6� �Q�� µ � �c��

16.� � �� ¥ � �� À �Q�� µ � � �c��

17.

��#� ��#� �� À � �� c�� µµ � � � �{��A�18.

� � �� FýJ� � �� FýJ� � �� FýJ� � �� Fýá� �Q�� µ �FýJ� �Q��19.

� � �� ù � �� ù �Q�� µ ù �c�A�� , whereú ç �.

20. Write a definition for ¹»º»¼B½ « Ö ±�þsimilar to the ÿ ¿��

definitionfor ¹»º ¼è ½ � � ¦¸� § ±dþin Section2.2.Insteadof � ,

youwill need� , avalueof µ .

21. Show thatif Þ Ô and Þ�� convergeandif � is acon-

stant,then Þ ¦ Ô � � § , Þ ¦ Ô � � § , and Þ �Ô

con-verge.

22. Let � be a positive integer. Show that if

Ô ± � forµ�� , then Þ Ô and Þ � eitherbothconverge,or

bothdiverge.In otherwords,changingafinite numberof

termsin a seriesdoesnot changewhetheror not it con-verges.

23. Show thatif Þ Ô converges,then ¹»º»¼6½ «

Ô ± � ´ [Hint:

Consider¹»º ¼ B½ « ¦(Ö � Ö ô © § , whereÖ is the µ�¶¸· par-tial sum.]

24. Show that if «¬ 6® ©Ô

divergesand ��± �, then «¬B® © �

Ô diverges.

25. Considerthe following grouping of terms in the har-monicseries:� � ��;� �� 8� ��#� �� À � ��;� �¥ � ��A� �{��A�� A� �;�L��(a) Show thatthesumof eachgroupof fractionsis more

than

� Æ��.

(b) Explain why this shows that the harmonicseriesdoesnotconverge.

26. Estimatethe sumof the first

�A�� ¿ �6�� termsof the har-monicseries, ©(ªMª�� ªMªMª¬ ��® © �� ¿to the closestinteger. [Hint: Use left- and right-handsumsof the function

� ¦¸� § ± � Æ � on the interval from�to

��6� ¿ ��6� with �L� ± �.]

27. Althoughtheharmonicseriesdoesnotconverge,thepar-tial sumsgrow very, very slowly. Take a right-handsumof� ¦¸� § ± � Æ � with �L� ± �

on the interval � � ¿ µ�� toshow that �� c�� µ ël¹»êHµ ´

9.3 TESTS FOR CONVERGENCE 439

If a computercouldadda million termsof theharmonicserieseachsecond,estimatewhatthesumwouldbeafteroneyear.

28. In thisproblem,youwill justify theintegraltest.Suppose� � �and

� ¦¸� § is adecreasingpositive function,definedfor all ��

, with� ¦¸µ § ± Ô

for all µ .

(a) Supposeâ «� � ¦¸� § é � converges.By drawing rect-

anglesunderthecurve, show that Þ Ô converges.

[Hint: SeeExample4 onpage436.]

(b) Supposethat â «� � ¦¸� § é � diverges. By drawing

rectanglesabove the curve, show that Þ Ô di-

verges.[Hint:SeeExample3 onpage436.]

9.3 TESTS FOR CONVERGENCE

Comparison of SeriesIn Section7.8, we comparedtwo integralsto decidewhetheran improperintegral converged.InTheorem9.3wecomparedanintegralandaseries.Now wecomparetwo series.

Theorem 9.4: ComparisonTest

Suppose Ó ^ "ìÓ Ü " for all $ �Ï If Þ Ü " converges,then Þ ^ " converges.Ï If Þ ^ " diverges,then Þ Ü " diverges.

Since9" Ó Ü " , theplot of the ^9" liesundertheplot of the Ü " . (SeeFigure9.6.)Thecomparisontestsaysthatif thetotalareafor Þ Ü " is finite, thenthetotalareafor Þ ^9" is finite also.If thetotalareafor Þ ^1" is notfinite, thenneitheris thetotalareafor Þ Ü " .

� �}� � � µÔ © Ô �

Ô� Ô � Ô

� © � � � � � � � � �Ô . . . . . .

Figure 9.6: Each

Ô is representedby theareaof adarkrectangle,andeach� by adarkplusa light rectangle

Example 1 Usethecomparisontestto determinewhethertheseries

r¬"�Ì � � w¡$ � converges.

Solution For $ × �, weknow that $ � Ó $ � , so Ó �$ � Ó �$ � �

Thus, every term in the series Þ r"�Ì � � w¡$ � is less than or equal to the correspondingterm inÞ r"�Ì � � wB$ � . Sincewe saw that Þ r"�Ì � � w¡$ � convergesfrom page436,we know that Þ r"�Ì � � wB$ �converges.


Example 2 Usethecomparisontestto show thatif k{ò�k �

, then Þ r"�Ì � � w¡$:ñ diverges.

Solution Sinceò�k �and $ × �

, wehave $ ñüÓ $ , so� wB$ ñ�× � wB$ . TheharmonicseriesÞ � wB$ diverges,soÞ � wB$:ñ divergesalso.

Example 3 Decidewhetherthefollowing seriesconverge: (a)r¬"�Ì � $ P �$ � �� (b)

r¬"�Ì � 7 $ � �,�+ $ � P � .

Solution (a) Sincetheconvergenceis determinedby thebehavior of thetermsfor large $ , weobservethat$ P �$ � �� Y $$ � ) �$ � as $yYzZ �SinceÞ � wB$ � converges,weguessthat Þ . $ P �B0 w . $ � �l��0

converges.To confirmthis,weusethecomparisontest.Sinceafractionincreasesif its numeratoris madelargeror its denominatoris madesmaller, wehave Ó $ P �$ � �� Ó $$ � ) �$ � for all $ × ��Thus,theseriesÞ . $ P �60 w . $ � ��0

convergesby comparisonwith Þ � wB$ � .(b) First,weobservethat 7 $ � ��+ $ � P � Y 7 $ �+ $ � ) �$ as $�YzZ �

Since Þ � w¡$ diverges,sodoesÞ � wB$ , andweguessthat Þ . 7 $ � ��60 w . + $ � P �60diverges.To

confirm this, we usethe comparisontest.Sincea fraction decreasesif its numeratoris madesmalleror its denominatoris madelarger, wehave Ó 7 $ �+ $ � Ó 7 $ � ��+ $ � P � �so Ó �$ Ó 7 $ � ��+ $ � P � �Thus,theseriesÞ . 7 $ � ��B0 w . + $ � P �60

divergesby comparisonwith Þ � wB$ .

Series of Both Positive and Negative TermsIf Þ ^ " hasboth positive and negative terms,then its plot hasrectangleslying both above andbelow the horizontalaxis. SeeFigure9.7. The total areaof the rectanglesis no longerequaltoÞ ^ " . However, it is still true that if the total areaof the rectanglesabove andbelow the axis isfinite, thentheseriesconverges.Theareaof the $&%(' rectangleis i ^1"8i , sowehave:

Theorem 9.5: If Þ��! "� converges,then sodoes Þ#�! .

Problem44showshow to provethis result.


µÔ Ô Ô �Ô�Ô�

Ô ©. . .

. . .

Figure 9.7: Representingaserieswithpositiveandnegative terms

Example 4 Explainhow weknow thatthefollowing seriesconvergesr¬"�Ì � . P �60 " ` �$ � ) � P �� P ��Solution Writing ^ " ) . P �B0 "9` � wB$ � , wehavei ^ " i )%$$$$$

. P �60 " ` �$ � $$$$$ ) �$ � �FromExample4 onpage436,weknow that Þ � w¡$ � converges,so Þ . P �60 "9` � w¡$ � converges.

Comparison with a Geometric Series: The Ratio TestWe now develop a test for convergenceby comparisonwith a geometricseries.Recall that it iseasyto tell if a geometricseriesconverges.If _ is thecommonratio betweensuccessive termsof ageometricseries,thentheseriesconvergesif i _Ki:k �

anddivergesif i _Ki × �. Thefollowing testis

obtainedby comparingtheseriesÞ i ^ " i with a geometricseriesas $�YÑZ .

Theorem 9.6: The Ratio Test

For a seriesÞ ^ " , supposethesequenceof ratios i ^ "�ã � i w�i ^ " i hasa limit:montp"�q�r i ^ "�ã � ii ^1"8i ) Î �Ï If Î k �, then Þ ^ " converges.Ï If Î x �, or if Î is infinite,3 then Þ ^ " diverges.Ï If Î ) � � thetestdoesnot tell usanythingabouttheconvergenceof Þ ^ " .

3Thatis, thesequence& ' )( © & *)& ' & grows withoutbound.


Proof Thebasicideais to comparetheserieswith a geometricseries.SupposeÎàk �

. Let _ beanumberbetweenÎ and�, sothat Îàkà_yk �

. Sincemontp"�q�r i ^ "�ã � ii ^1"8i ) Î �for all sufficiently large $ , sayfor all $ × Ý , wehavei ^ "�ã � ii ^ " i kà_ �Then, i ^ ßAã � i9kRi ^ ß i _ �i ^ ßAã � i9kRi ^ ßAã � i _ykVi ^ ß i _ � �i ^ ßAã � i9kRi ^ ßAã � i _ykVi ^ ß i _ � �andsoon.Thus,for a sufficiently large $ , wehavei ^ ß i � i ^ ßAã � i � i ^ ßAã � i � i ^ ß�ã � i �� kRi ^ ß i � i ^ ß i _ � i ^ ß i _ � � i ^ ß i _ � ��Thus, i ^ ß i � i ^ ßAã � i � i ^ ß�ã � i � i ^ ßAã � i �g��

convergesby comparisonwith theconvergentgeometricseriesi ^ ß i � i ^ ß i _ � i ^ ß i _ � � i ^ ß i _ � ��

. Hence,Þ i ^ " i converges,so Þ ^ " converges.If Îàx �

, thenwechoose_ sothat� kà_�kàÎ . Then,for sufficiently large $ , say $ ×,+ ,i ^9"�ã � i9xÉi ^9"8i _yxRi ^.-üi �

sothesequencei ^ - i � i ^ - ã � i � i ^ -�ã � i � �� , is increasing.Thus,

mtnop"�qsr ^ " b) , so Þ ^ " diverges(by

Theorem9.2,property3).

Example 5 Show thatthefollowing seriesconverges:4r¬"�Ì � �$0/ ) �~� �+ / � �� / ��Solution Since " ) � w¡$0/ and ^ "�ã � ) � w . $ ��B0 / , wehavei ^ "�ã � ii ^ " i ) � w . $ ��60 /� wB$0/ ) $0/. $ ��60 / ) $ . $ P �B0�. $ PQ+ 0j�� + ��. $ ��B0 $ . $ P �60j�� + ��

We cancel$ . $ P �B0�. $ PQ+ 0j�� + �� , givingmtnop"�qsr i ^ "�ã � ii ^9"8i ) montp"�q�r $0/. $ ��B0 / ) mtnop"�qsr �$ �� ) ��Becausethelimit is 0, which is lessthan

�, theratio testtellsusthat

r¬"�Ì � � wB$0/ converges.

4Wedefine132 (readtwo factorial) to be 1547698:1 . Similarly, ;32<8=;54>154?698=@ and A�278:AB4?CDAFEG6IHJ4I4K4L15476 .


Example 6 Whatdoestheratio testtell usabouttheconvergenceof thefollowing two series?r¬"�Ì � �$ andr¬"�Ì � . P �B0 "9` �$ �

Solution Becausei . P �B0 " i ) � � in bothcaseswehave

monop "�qsrÉi ^9"�ã � w¡^9";i ) mtnop "�qsrà$�w . $ �e�60 ) �. Thus,

theratio testdoesnot tell usanythingabouttheconvergenceof eitherseries.In fact,thefirst seriesis theharmonicseries,whichdiverges.Example7 will show thatthesecondseriesconverges.

Alternating SeriesA seriesis calledanalternating series if thetermsalternatein sign.For example,r¬"�Ì � . P �B0 "9` �$ ) � P �+ � �� P �� B� . P �60 " ` �$ �,��Theconvergenceof analternatingseriescanoftenby determinedusingthefollowing test:

Theorem 9.7: Alter nating SeriesTest

A seriesof theformr¬"�Ì � . P �60 "9` � ^ Í ) ^ � P ^ � � ^ � P ^ ��6�,. P �60 "9` � ^ " ��convergesif k ^ "�ã � ke^ " for all $ and

montp"�q�r ^ " ) ��Althoughwe do not prove this result,we canseewhy it is reasonable.The first partial sum,a � ) ^ � , is positive.Thesecond,a � ) ^ � P ^ � , is still positive,since ^ � kR^ � , but a � is smaller

than a � . (SeeFigure9.8.)Thenext sum, a � ) ^ ��P ^ � � ^ � , is greaterthan a � but smallerthana � . Thepartialsumsoscillatebackandforth, andsincethedistancebetweenthemtendsto 0, theyeventuallyconverge.

� Ö � Ö � Ö � Ö ©å Må MÔ�Ô �Ô �Ô © �

Figure 9.8: Partial sums,Ö © , Ö � , Ö � , Ö � of analternatingseries

Example 7 Show thatthealternatingharmonicseriesconvergesr¬"�Ì � . P �60 "9` �$ �


Solution We have ^9" ) � w¡$ and ^1"�ã �*) � w . $ ��60. Thus,^ "�ã � ) �$ �� k �$ ) ^ " for all $ , and

montp"�qsr � w¡$ ) ��Thus,thehypothesisof Theorem9.7is satisfied,sothealternatingharmonicseriesconverges.

Supposea is thesumof thealternatingseries,so a ) monop "�qsr a;" . Then a is trappedbetweenany two consecutivepartialsums,say a � and a or a and a;C soa � k a k �� k�a�k �� k�a � kàa � �Thus,theerrorin using a;" to approximatethetruesum a is lessthanthedistancefrom a;" to a8"�ã � ,which is ^9"�ã � . Statedsymbolically, wehavethefollowing result:

Theorem 9.8: Err or Boundsfor Alter nating Series

Let a " ) "¬ Í Ì � . P �60 Í ` � ^ Í bethe $&%(' partialsumof analternatingseriesandlet a ) montp"�q�r a " .

Supposethat k ^1"�ã � kà^9" for all $ and

mtnop "�qsr ^9" ) . Theni a P a " i kà^ "�ã � �

Thus,theerrorin using a " to approximatea is lessthanthemagnitudeof thefirst termof theserieswhich is omittedin theapproximation.

Example 8 Estimatetheerror in approximatingthesumof thealternatingharmonicseries

r¬"�Ì � . P �60 "9` � wB$ by

thesumof thefirst nineterms.

Solution Theninthpartialsumis givenbya G ) � P �+ � �� P ��6� �� ) �� -�7 ��Thefirst termomittedis P � w �� , with magnitude

��o�. By Theorem9.8,weknow thatthetruevalue

of thesumdiffersfrom �� -�7 �� by lessthan

��o�.


Decideif the statementsin Problems1–13aretrue or false.Giveanexplanationfor youranswer.

1. If

� û Ô û � and Þ Ô converges,then Þ � con-

verges.

2. If

� û Ô û � and Þ Ô diverges,the Þ � diverges.

3. If � û Ô û �and Þ � converges,then Þ Ô

con-verges.

4. If Þ Ô converges,then ÞON

Ô N converges.


5. If ÞPNÔ � � N converges,then ÞQN

Ô N and ÞON � N con-verge.

6. If Þ Ô converges,then ¹»º»¼6½ « N

Ô 3( © N Æ N Ô N �± �.

7. «¬B® ª ¦ �*� § õ�ö6÷�¦¸ø�µ § is analternatingseries.

8. «¬B® © ¦ � �Q¦ �*� § § is aseriesof nonnegative terms.

9. Theseries «¬ B® ª ¦ �*� § � converges.

10. Theseries «¬ B® © �SR ô ©LT�U converges.

11. If Þ Ô converges,then Þ ¦ �*� § Ô converges.

12. If ÞONÔ N converges,then Þ ¦ �*� § N Ô N converges.

13. To find the sum of the alternatingharmonic seris towithin

� ´ �� of thetruevalue,weshouldsumthefirst 100terms.

Use the comparisontest to confirm the statementsin Prob-lems1–2.

14. «¬B® © �µ � converges,so «¬ 6® © �µ � � � converges.

15. «¬B® � �µ diverges,so «¬B® � �µ � � diverges.

16. «¬B® © �µ � converges,so «¬ 6® © ó ô µ � converges.

Use the comparisontest to determinewhetherthe seriesinProblems17–24converge.

17. «¬B® © �µ �18. «¬B® � �¹ êHµ19. «¬B® © µµs� �

20. «¬ 6® © �� 21. «¬ 6® © �µ � �ló 22. «¬ 6® © � ô ¦¸µ#� � §¦¸µ#� �6§23. «¬ 6® © µ �µ � � �24. «¬ 6® © � � �µ � �g�Usetheratiotestto decidewhichof theseriesin Problems25–29 convergeandwhichdiverge.

25. «¬ 6® © �¦ � µ §IV26. «¬ 6® © ¦¸µ V § �¦ � µ §IV27. «¬ 6® ª � µ � � �28. «¬ 6® © �µ�ó 29. «¬ 6® © µ ô Usethealternatingseriestestto decidewhichof theseriesinProblems30–32converge.

30. «¬ 6® © ¦ �*� § ô ©� µs� �31. «¬ 6® © ¦ �*� § ô ©Â µ32. «¬ 6® © ¦ �*� § ô © µ Vµ

Problems

Use a computeror calculatorto investigatethe behavior ofthe partial sumsof the alternatingseriesin Problems33–35.Which onesconverge?Confirm convergenceusingthe alter-natingseriestest.If aseriesconverges,estimateits sum.

33.�@�ì� ´ � � � ´ ��~� � ´ �� Q��Q¦ �*� § �A� ô �Q��

34.�@� � � � � � � � �c��Q¦ �*� § ¦¸µ#� � § �c��

35.�@� �� V � ��SV � ��SV �Q��Q¦ �*� § �µ V �Q��

Determinewhichof theseriesin Problems5–11converge.

36. «¬ 6® © W ��W� � �µ �37. «¬ 6® © µ#� � µ �


38. «¬B® © ¦ �*� § ô ©Â � µ �l�39. «¬B® © ¦ �*� § ô © � µ �40. «¬B® © �Å µ8¦¸µs� � § ¦¸µ#� �6§41. Is thefollowing argumenttrueor false?Give reasonsfor

youranswer.

Considerthe infinite series «¬ B® � �µ8¦¸µ �l� § ´ Since�µ8¦¸µ �l� § ± �µ �l� � �µ wecanwrite thisseriesas«¬ B® � �µ �l� � «¬ B® � �µ ´For the first series

Ô ± � Æ ¦¸µ �d� §. Since µ �d� ë,µ

wehave

� Æ ¦¸µ �l� § ç � Æ µ andsothisseriesdivergesby

comparisonwith the divergent harmonicseries «¬B® � �µ .

Thesecondseriesis thedivergentharmonicseries.Sincebothseriesdiverge,theirdifferencealsodiverges.

42. Show that if Þ NÔ N converges,then Þ ¦ �*� § Ô con-

verges.

43. (a) Show thatif µ is even,�6� �� ;�L�� µ ± �� ;�L�� ¦¸µ �l� § �Aµ ´(b) Show that

�� ¨� �� c�A�� converges.

(c) Use the resultsof parts (a) and (b) to show that�¾� �� c��A� converges.

44. Let Þ Ô beaseriessuchthat ÞON

Ô N converges.In thisproblem,youwill show that Þ Ô

converges.

(a) Defineanew seriesÞ � , asfollows,andshow thatit converges:

� ±YXÔ

if

Ô � ��if

Ô ë � ´(b) Defineanew seriesÞ � , asfollows,andshow that

it converges:� ±YX � if

Ô � �� Ô if

Ô ë � ´(c) Use Þ � and Þ � to show that Þ Ô

converges.

9.4 POWER SERIES

In Section9.1 we saw that the geometricseriesÞ ^1_ " convergesfor P � ku_,k �anddiverges

otherwise.This sectionstudiesthe convergenceof moregeneralseriesconstructedfrom powers.Chapter??showshow suchpowerseriesareusedto approximatefunctionssuchas Ð)Z , [ noä _ , \7]^[ _ ,and

moä _ .

A power seriesabout_ ) ^ is a sumof constantstimespowersof. _ P ^ 0 :_ E � _ � . _ P ^ 0£� _ � . _ P ^ 0 � ��B� _ " . _ P ^ 0 " �� ) r¬"�Ì E _ " . _ P ^ 0 " �

We think of ^ asa constant.For any fixed _ , the power seriesÞ _ " . _ P ^ 0 " is a seriesofnumberslike thoseconsideredin Section9.2.To investigatetheconvergenceof a powerseries,weconsiderthepartialsums,whichin thiscasearethepolynomialsa;" . _ 0 ) _ E � _ � . _ P ^ 0�� _ � . _ P^ 0 � ��6� _ " . _ P ^ 0 " . As before,weconsiderthesequence5a E . _ 0 � a � . _ 0 � a � . _ 0 � �� a;" . _ 0 � ��

For a fixed valueof _ , if this sequenceof partial sumsconvergesto a limit Î , that is, ifmonop"�qsr a " . _ 0 ) Î , thenwesaythatthepowerseriesconvergesto Î for thisvalueof _ .

A powerseriesmayconvergefor somevaluesof _ andnot for others.5Herewehavechosento call thefirst termin thesequence ª CbaJH ratherthan ` © C�aJH to correspondto thepowerof C�a�Ec'3H .

9.4 POWER SERIES 447

Numerical and Graphical View of ConvergenceAs anexample,considertheseries. _ P �60 P . _ P �B0 �+ � . _ P �60 �� P . _ P �60 � �,��6�,. P �B0 "9` � . _ P �B0 "$ ��To investigatetheconvergenceof thisseries,welook atthesequenceof partialsumsgraphedin Fig-ure9.9.Thegraphsuggeststhatthepartialsumsconvergefor _ in theinterval

./ � + 0 . In Examples2and 5, we show that theseriesconvergefor

k,_ Ó + . This is calledthe interval of convergenceof thisseries.

At _ ) �� , which is insidetheinterval, theseriesappearsto convergequiterapidly:a C .F�� 10 ) �� 7 ��*�� a N .F�� 10 ) �� 7�- �*��a;� .F�� 10 ) �� 7 �� *�� a I .F�� 10 ) �� 7 � - ��

Table9.1 shows the resultsof using _ ) �� and _ )�+ � � in the power series.For _ ) ��

,which is insidethe interval of convergencebut closeto anendpoint,theseriesconverges,thoughratherslowly. For _ ) + � � , which is outsidethe interval of convergence,the seriesdiverges:thelarger thevalueof $ , themorewildly theseriesoscillates.In fact, thecontribution of the twenty-fifth termis about28; thecontributionof thehundredthtermis about P�+ � - � � � � � � � . Figure9.9showstheinterval of convergenceandthepartialsums.

� ´ ¥ � ´ � �Ö ©M© ¦¸� § ÖedB¦¸� §

Ö ¦¸� §Ö © � ¦¸� §

� ± ��

Må Intervalof

convergence�Figure 9.9: Partialsumsfor seriesinExample2 convergefor

� ël�üë �

Table 9.1 Partial sums for series inExample 2 with _ ) ��

inside intervalof convergence and _ )D+ � � outsideµ Ö ¦ � ´ ¥ § µ Ö ¦ � ´ �B§

2 0.495 2 0.455

5 0.69207 5 1.21589

8 0.61802 8 0.28817

11 0.65473 11 1.71710

14 0.63440 14

�¾� ´ À � À ��Notice that the interval of convergence,

k _ Ó + , is centeredon _ ) �. Sincethe interval

extendsoneunit oneitherside,wesaytheradius of convergence of thisseriesis 1.

Inter vals of ConvergenceEachpower seriesfalls into oneof thethreefollowing cases,characterizedby its radius of conver-gence, f .Ï Theseriesconvergesonly for _ ) ^ ; theradius of convergenceis definedto be f )

.Ï The seriesconvergesfor all valuesof _ ; the radius of convergenceis definedto bef ) Z .Ï Thereis a positive number f , called the radius of convergence, suchthat the seriesconvergesfor i _ P ^;i9kgf anddivergesfor i _ P ^;i9xhf . SeeFigure9.10.

448 Chapter Nine SERIES Ô �=i Ô Ô � iSeriesdiverges

Må Interval of convergenceSeries

diverges �Må i

Figure 9.10: Radiusof convergence,

i, determinesaninterval, centeredat � ± Ô

, inwhich theseriesconverges

Therearesomeserieswhoseradiusof convergencewe alreadyknow. For example,the geo-metricseries �~� _ � _ � ��¡� _ " ��convergesfor i _Ki k �

anddivergesfor i _ji × �, so its radiusof convergenceis 1. Similarly, the

series �~� _ � � _ � � � ��¡� _ � � " ��convergesfor i _;w � i9k �

anddivergesfor i _;w � i × �, soits radiusof convergenceis 3.

Thenext theoremgivesa methodof computingtheradiusof convergencefor many series.To

find the valuesof _ for which the power series

r¬"�Ì E _ " . _ P ^ 0 " converges,we usethe ratio test.

Writing ^9" ) _ " . _ P ^ 0 " andassuming_ "gb)

and _Qb) ^ , wehavemontp"�q�r i ^ "�ã � ii ^9"8i ) monop"�qsr i _ "�ã � . _ P ^ 0 "�ã � ii _ " . _ P ^ 0 " i ) monop"�qsr i _ "�ã � ioi _ P ^;ii _ "£i ) i _ P ^;i montp"�qsr i _ "�ã � ii _ "£i �Case1. Supposei ^ "�ã � i w�i ^ " i is unbounded.Thenthe ratio testshows that the power seriescon-vergesonly for _ ) ^ . Theradiusof convergenceis f )

.Case2. Suppose

montp"�q�r i ^ "�ã � i w�i ^ " i ) . Thentheratio testshows that thepower seriesconverges

for all _ . Theradiusof convergenceis f ) Z .Case3. Suppose

mtnop"�qsr i ^9"�ã � i w�i ^9"8i ) Òài _ P ^&i , where

mtnop"�qsr i _ "�ã � i w�i _ "8i ) Ò . In Case1, Òdoesnot exist; in Case2, Ò )

. Thus,we canassumeÒ existsand Ò]b) , andwe candefinef ) � w�Ò . Thenwehave montp"�qsr i ^9"�ã � ii ^ " i ) Òei _ P ^;i ) i _ P ^;if �

sotheratio testtellsusthatthepowerseries:Ï Convergesfori _ P ^;if k �

; thatis, for i _ P ^;i9kgfÏ Divergesfori _ P ^;if x �

; thatis, for i _ P ^;i9xjf .

Theresultsaresummarizedin thefollowing theorem.

Theorem 9.9: Method for Computing Radiusof Convergence

To calculatetheradiusof convergence,f , for thepowerseries

r¬"�Ì E _ " . _ P ^ 0 " , usetheratio

testwith ^ " ) _ " . _ P ^ 0 " .Ï If

mtnop"�qsr i ^ "�ã � i w�i ^ " i is unbounded,then f ) .Ï If

mtnop"�qsr i ^ "�ã � i w�i ^ " i ) , then f ) Z .Ï If

mtnop"�qsr i ^9"�ã � i w�i ^9"8i ) Òei _ P ^;i , whereÒ is finite andnonzero,when f ) � w¡Ò .


Notethattheratio testdoesnot tell usanythingif

montp "�qsrRi ^9"�ã � i w�i ^9"8i fails to exist,whichcanoccurif someof the

_ " sarezero.A proof that a power serieshasa radiusof convergenceandof Theorem9.9 is given in the

onlinetheorysupplement.To understandthesefactsinformally, we canthink of a power seriesasbeinglike a geometricserieswhosecoefficientsvary from termto term.Theradiusof convergencedependsonthebehaviour of thecoefficients:if thereareconstants

_and Ò suchthatfor largerand

larger $ , i _ " i \ _ Ò " �thenit is plausiblethat Þ _ "9_ " and Þ _ Ò " _ " ) Þ _ . Òy_ 0 " convergeor divergetogether. ThegeometricseriesÞ _ . Òl_ 0 " convergesfor i Òl_ji9k �

, thatis, for i _ji�k � w¡Ò . We havei ^ "�ã � ii ^ " i ) i _ "�ã � ioi . _ P ^ 0 "�ã � ii _ " ioi . _ P ^ 0 " i \ _ Ò "�ã � i . _ P ^ 0 "�ã � i_ Ò " i . _ P ^ 0 " i ) Òài _ P ^;i �Example 1 Show thatthefollowing powerseriesconvergesfor all _ :�~� _ � _ �+ / � _ �� / ��B� _ "$0/ ��Solution Because

_ " ) � wB$0/ , noneof the_ " sarezeroandwecanusetheratio test:mtnop"�qsr i ^ "�ã � ii ^9"£i ) i _ji mtnop"�qsr i _ "�ã � ii _ "£i ) i _ji mtnop"�qsr � w . $ ��60 /� wB$0/ ) i _ji mtnop"�qsr $0/. $ ��60 / ) i _ji mtnop"�qsr �$ �� ) ��

Thisgives f ) Z , sotheseriesconvergesfor all _ . We seein Chapter?? thatit convergesto Ð)Z .Example 2 Determinetheradiusof convergenceof theseries. _ P �60 P . _ P �B0 �+ � . _ P �60 �� P . _ P �60� �,��6�,. P �B0 "9` � . _ P �B0 "$ ��

Whatdoesthis tell usabouttheinterval of convergenceof thisseries?

Solution Thegeneraltermof theseriesis. _ P �B0 " wB$ if $ is oddand P . _ P �B0 " wB$ if $ is even,so

_ " ). P �B0 "9` � w¡$ , andwecanusetheratio test.We havemtnop"�qsr i ^ "�ã � ii ^9"£i ) i _ P � i montp"�q�r i _ "�ã � ii _ "�i ) i _ P � i montp"�qsr i . P �60 " w . $ ��60 ii . P �60 "9` � wB$¾i ) i _ P � i mtnop"�qsr $$ �� ) i _ P � i �Theradiusof convergenceis f ) �

. Thepower seriesconvergesfor i _ P � i:k �anddivergesfori _ P � iHx �

, so the seriesconvergesfor k�_Dk + . Notice that the radiusof convergencedoes

not tell uswhathappensat theendpoints,_ ) and _ )�+ . We seein Chapter?? that theseries

convergesto

moä _ for _ in its interval of convergence.

Theratio testrequires

montp"�q#Ì�r i ^9"�ã � i w�i ^1";i to exist for ^1" ) _ " . _ P ^ 0 " . Whathappensif some

of thecoefficients_ " arezero?Thenweusethefactthataninfinite seriescanbewritten in several

waysandpick onein which thetermsarenonzero.For example,we think of thepowerseries_ P _ �� / � _ C- / P _ N� / ��6�,. P �B0 "9` � _ � "9` �. + $ P �60 / ��asthe serieswith ^ � ) _ and ^ � )ÚP _ � w � / �� , so ^ " ) . P �B0 "9` � _ � "9` � w . + $ P �B0 / . With thischoiceof ^ " , all ^ " b)

, sowecanusetheratio test.6

6Wedonot take ' © 8:a.kL' � 8=l3kL' � 8mEna � *o;32pkL' � 8=l3kKq>qKq becausethen rpsptB½ « & ' )( © & *)& ' & doesnot exist.


Example 3 Find theradiusandinterval of convergenceof theseries_ P _ �� / � _ C- / P _ N� / ��6�,. P �B0 "9` � _ � "9` �. + $ P �60 / ��Solution We take ^9" ) . P �B0 "9` � _ � "9` �. + $ P �60 / �

sothat,replacing$ by $ ��, wehave^1"�ã �*) . P �60�u "�ã �>v ` � _ � u "�ã �Kv ` �. + . $ ��60 P �60 / ) . P �60 " _ � "�ã �. + $ ��B0 / �

Thus,i ^9"�ã � ii ^ " i ) $$$ . P �60 " Z3w U3x^yu � "�ã �>vLz $$$$$$ . P �60 "9` � Z w U|{^yu � " ` �>vLz $$$ ) $$$$. P �60 " _ � "�ã � . + $ P �B0 /. P �B0 "9` � _ � "9` � . + $ ��60 / $$$$ ) $$$$

. P �60 _ �. + $ ��60 + $ $$$$ ) _ �. + $ ��60 + $ �Because montp"�qsr i ^ "�ã � ii ^ " i ) mtnop"�qsr _ �. + $ ��B0 + $ ) �wehave Î ) montp "�qsr i ^ "�ã � i w�i ^ " i ) k �

for all _ � Thus,theratio testguaranteesthatthepowerseriesconvergesfor every _ . Theradiusof convergenceis infinite andthe interval of convergenceis all _ . We seein Chapter?? thattheseriesconvergesto [ noä _ .

Example 4 Find theradiusandinterval of convergence�~� + � _ � � + _ � + � _ � ��¡� + � " _ � " ��Solution If wetake ^9" ),+ " _ " for $ evenand ^9" )

for $ odd,

montp"�qsr i ^9"�ã � i w�i ^1"8i doesnotexist.Therefore,

for thisserieswe take ^ " )D+ � " _ � " �sothat,replacing$ by $ ��

, wehave^ "�ã � )D+ � u "�ã �>v _ � u "�ã �Kv ),+ � "�ã � _ � "�ã � �Thus, i ^1"�ã � ii ^ " i ) $$ + � "�ã � _ � "�ã � $$i + � " _ � " i ) $$ + � _ � $$ ) � _ � �Because montp"�q�r i ^9"�ã � ii ^ " i ) � _ � �wehave Î ) montp i ^1"�ã � i w�i ^9"8i ) � _ � . Thus,theratio testguaranteesthatthepowerseriesconvergesif� _ � k � � thatis, if i _ji9k �� . Theradiusof convergenceis

�� . Theseriesconvergesfor P �� ke_�k ��anddivergesfor _lk �� or _yx P �� . At _ )} �� , all thetermsin theseriesare1,sotheseriesdiverges(by Theorem9.2Property3). Thus,theinterval of convergenceis P �� kà_�k �� .


What Happens at the Endpoints of the Inter val of Convergence?The ratio testdoesnot tell us whethera power seriesconvergesat the endpointsof its interval ofconvergence,_ ) ^ } f . Thereis nosimpletheoremthatanswersthisquestion.Sincesubstituting_ ) ^ } f convertsthepower seriesto a seriesof numbers,the testsin Sections9.2 and9.3 areoftenuseful.SeeExamples4 and 5.

Example 5 Determinetheinterval of convergenceof theseries. _ P �60 P . _ P �B0 �+ � . _ P �60 �� P . _ P �60� �,��6�,. P �B0 "9` � . _ P �B0 "$ ��Solution In Example2 on page449,we showedthatthis serieshas f ) �

; it convergesfor k,_Qk + and

divergesfor _�k or _�x + . We needto determinewhetherit convergesat the endpointsof the

interval of convergence,_ ) and _ ),+ . At _ ),+ , wehave theseries� P �+ � �� P �� B� . P �60 " ` �$ ��

This is analternatingserieswith ^9" ) � w . $ �,�60, soby thealternatingseriestest(Theorem??), it

converges.At _ ) , wehavetheseriesP � P �+ P �� P �� P �� P �$ P ��

This is thenegativeof theharmonicseries,soit diverges.Therefore,theinterval of convergenceis k�_ Ó + . Theright endpointis includedandtheleft endpointis not.


Whichof theseriesin Problems1–4arepowerseries?

1. � � � � �� ¢ � � ©(ª �� © �� 2.

�� ¨��3.

� �ü�¾�|¦¸� �� § � ��¦¸� ��6§ � �c¦¸� � �6§ � �{�A�� 4. �e~8�y�� Find anexpressionfor thegeneraltermof theseriesin Prob-lems 5–10.Give the startingvalueof the index ( µ or � forexample).

5.

��L� � � �� |V>� � � � � � � �� SV�� Q��6. ú9�� ú&¦ ú �l� §�SV � � � ú&¦ ú �g� § ¦ ú � ��§�JV � � �Q��A�7.

�@� ¦¸� �g� § ��SV � ¦¸� �g� § �� V � ¦¸� �l� § ¢� V �Q��8. ¦¸� �� § � � ¦¸� �y� § �SV � ¦¸� �l� § ~� V � ¦¸� �l� §�� V �g��

9.� � Ô� � ¦¸� � Ô § �� SV � ¦¸� � Ô § �� JV � ¦¸� � Ô § �� V �c��

10.� ¦¸� � ��§ � � � ¦¸� � �6§ � � ¦¸�� 6§ ~�|V � � ¦¸�� § ��SV �y��A�

Find theradiusof convergenceof theseriesin Problems12–15.

11. «¬ 6® ª ¦ � � § 12. «¬ 6® ª µ�� ¿>� ç �13. «¬ 6® ª ¦¸µ � � § � � ��µ14. «¬ 6® ª � ¦¸� �y� § µ


Use the ratio test to find the radius of convergenceof thepower seriesin Problems15–21.

15. �j� � � � ��¥�� 1��6� � �Q�� 16. � � � �� ¥ � �9�� 6� � �A��17.

� � � �L� � � ��SV � � � ��JV � �� V � �B� � �SV �c��18.

� � � � � �� À � � � �¥ � � � �� Q��19.

� � � �L� � V � �¦ �|V § � � � V � �¦ �JV § � � �JV � �¦ � V § � � �A� V � ¦ �|V § � �c��20.

� �L� �� À� � � � ¥�:� � � �6�� c��21. � � � �� ~À �Q��

Problems

22. (a) Determinetheradiusof convergenceof theseries� � � �� Q��Q¦ �*� § ô © � µ �c�� ´Whatdoesthis tell usabouttheconvergenceof thisseries?

(b) Investigateconvergenceat theendpointsof the in-terval of convergenceof thisseries.

23. Show thattheseries «¬B® © ¦ � � § µ convergesfor N � N ë � Æ6�.

Investigatewhethertheseriesconvergesfor � ± � Æ6�and� ± �*� Æ��

.

24. For constantú , find theradiusof convergenceof thebi-nomialpower series:7� ��ú9�� ú&¦ ú �g� § � ��SV � ú:¦ ú �g� § ¦ ú � ��§ � ��JV �{�A�� ´

25. Show thatif � ª �� © ��:� � � � �:� � � � ��A� convergesfor N � N ë i

with

igiven by the radiusof convergence

test,thensodoes� © � � � � �L� � � � � � �c��A� .26. Supposethat thepower «¬B® ª � � convergeswhen � ±� �

anddivergeswhen � ±�À. Which of the following

aretrue,falseor notpossibleto determine?Give reasonsfor youranswers.

(a) Thepower seriesconvergeswhen � ± �A�.

(b) Thepower seriesconvergeswhen � ± �.

(c) Thepower seriesdivergeswhen � ± �.

(d) Thepower seriesdivergeswhen � ± �.

Decideif thestatementsin Problems27–31aretrueor false.Giveanexplanationfor youranswer.

27. Þ � ¦¸� �� § and Þ � � havethesameradii of con-vergence.

28. If Þ � � and Þ�� have thesameradii of conver-gence,thenthecoefficients, � and � , mustbeequal.

29. «¬ 6® © ¦¸� � µ § is apower series.

30. A geometricseriesis apowerseries.

31. If a power serieswith

i �± �,

i �±�� convergesat oneendpointof its interval of convergence,thenit convergesat theotherendpointalso.

REVIEW PROBLEMS FOR CHAPTER NINE

Exercises

Use the comparisontest to confirm the statementsin Prob-lems1–2.

1. «¬B® © �� converges,so «¬ 6® © � µ �� µ � � � � converges.

2. «¬B® © �µ diverges,so «¬B® © �µ*÷Mº»ê � µ diverges.

Usethecomparisontestto decidewhichof theseriesin Prob-lems3–4convergeandwhichdiverge.

3. «¬ 6® © � � µµ � � �4. «¬ 6® © � µ � �|µs� �µ � �

Determinewhichof theseriesin Problems5–11converge.

5. «¬ 6® ª � � � � 6. «¬ 6® © W �� µ � �

7For anexplanationof thename,seeSection10.2.

REVIEW PROBLEMS FOR CHAPTER NINE 453

7. «¬B® © �� ÷Mº»êHµ8. «¬B® © � ¦ � µ §IV9. «¬B® © ¦ � µ §IV¦¸µ V § �

10. «¬B® © ¦ �*� § ô ©Â µ#� �11. «¬ ��® © ¹»ê � � ��

Find theradiusof convergenceof theseriesin Problems12–15.

12. «¬ 6® ª µ�� 13. «¬ 6® ª ¦ � µ §IV � ¦¸µ V § �14. «¬ 6® ª ¦ � ��µ � § � 15. «¬ 6® ª � µ V � �

Problems

16. A repeatingdecimalcanalwaysbe expressedasa frac-tion. This problemshows how writing a repeatingdeci-malasageometricseriesenablesyouto find thefraction.Considerthedecimal

� ´ ��B��B�� ´A´�´�´(a) Usethefact that

� ´ ��6��6�� ´�´�´ ± � ´ �� ´ �� ´ �6��6� �� g�A�� to write 0.232323.. . asageometricseries.

(b) Usetheformulafor thesumof ageometricseriestoshow that

� ´ ��6��B�� ´�´�´ ± ��BÆ ¥6¥ .17. Cephalexin is anantibioticwith ahalf-life in thebodyof

0.9hours,takenin tabletsof 250mgevery six hours.

(a) Whatpercentageof thecephalexin in thebodyat thestartof asix-hourperiodis still thereat theend(as-sumingno tabletsaretakenduringthattime)?

(b) Write anexpressionfor ¯ © , ¯ � , ¯ � , ¯ � , where

mg, is the amountof cephalexin in the body rightafterthe µ ¶ · tabletis taken.

(c) Express � , ¯ � in closed-formandevaluatethem.(d) Write an expressionfor ¯ and put it in closed-

form.(e) If the patientkeepstaking the tablets,useyour an-

swerto part(d) to find thequantityof cephalexin inthebodyin thelongrun,right aftertakingapill.

18. AroundJanuary1, 1993,BarbraStreisandsigneda con-tractwith Sony Corporationfor Ê � million a yearfor 10years.Supposethefirst paymentwasmadeon thedayofsigningandthatall otherpaymentsaremadeon thefirstdayof theyear. Supposealsothatall paymentsaremadeinto a bankaccountearning

� Ëa year, compoundedan-

nually.

(a) How muchmoney wasin theaccount(i) On thenightof December31,1999?

(ii) On thedaythelastpaymentis made?(iii) Whatwasthe presentvalueof thecontracton

thedayit wassigned?

19. Onewayof valuingacompany is to calculatethepresentvalueof all its futureearnings.Supposeafarmexpectsto

sell Ê ��6�� worth of Christmastreesoncea yearforever,with the first salein the immediatefuture. What is thepresentvalue of this Christmastree business?Assumethattheinterestrateis 4

Ëperyear, compoundedcontin-

uously.

20. BeforeWorld War I, theBritish governmentissuedwhatare called consols, which pay the owner or his heirs afixed amountof money every yearforever. (Cartoonistsof thetimedescribedaristocratsliving off suchpaymentsas“pickled in consols.”) Whatshoulda personexpecttopayfor aconsolwhichpays � 10ayearforever?Assumethe first paymentis oneyear from the dateof purchaseandthat interestremains4% per year, compoundedan-nually. ( � denotespounds,theBritish unit of currency.)

Problems21–23areaboutbonds, which areissuedby a gov-ernmentto raisemoney. An individualwhobuysa$1000bondgivesthegovernment$1000andin returnreceivesafixedsumof money, calledthecoupon, every six monthsor every yearfor the life of the bond.At the time of the last coupon,theindividualalsogetsthe$1000,or principal back.

21. What is the presentvalueof a $1000bondwhich pays$50ayearfor 10years,startingoneyearfrom now? As-sumeinterestrateis 6% peryear, compoundedannually.

22. What is the presentvalueof a $1000bondwhich pays$50ayearfor 10years,startingoneyearfrom now? As-sumetheinterestrateis 4%peryear, compoundedannu-ally.

23. (a) What is the presentvalue of a $1000bond whichpays$50a yearfor 10 years,startingoneyearfromnow? Assumetheinterestrateis 5% peryear, com-poundedannually.

(b) Since$50is 5%of $1000,thisbondis oftencalleda5%bond.Whatdoesyouranswerto part(a) tell youabouttherelationshipbetweentheprincipalandthepresentvalueof this bondwhenthe interestrate is5%?


(c) If the interestrate is morethan5Ë

per year, com-poundedannually, which is larger: the principal orthevalueof thebond?Why doyouthink thebondisthendescribedastrading at discount?

(d) If the interestrate is less than 5Ë

per year, com-poundedannually, why is the bond describedastrading at a premium?

24. Supposethat the power «¬B® ª � ¦¸� � �6§ converges

when � ± �anddivergeswhen � ± �

. Whichof thefol-lowing aretrue,falseor not possibleto determine?Give

reasonsfor youranswers.

(a) Thepower seriesconvergeswhen � ±dÀ.

(b) Thepower seriesdivergeswhen � ± �.

(c) Thepower seriesconvergeswhen � ± � ´ � .(d) Thepower seriesdivergeswhen � ± �

.(e) Thepower seriesconvergeswhen � ± � �

.

25. (a) For a series Þ Ô , show that

� û Ô � NÔ N û� N Ô N .

(b) Usepart (a) to show that if ÞQNÔ N converges,thenÞ Ô

converges.

PROJECTS

Exercises

1. The Fly and the TrainIn anold puzzle,therearetwo trains,eachmovingat 10 km/hr towardoneanother. Initially thetrainsare30 kilometersapart.At thesamemomenta fly,whosevelocity is 20 km/hr, startsat onetrain andflies till it meetsthe other, then turnsaroundandfliesbacktill it meetsthefirst train,andsoon.

(a) How far has the fly traveled the first timeit turns around?The secondtime? The thirdtime?Thefourth time?

(b) How far hasthe fly traveled by the $&%(' timeit turnsaround?Write your answerin closed-form.

(c) Useyou answerto part (b) to decidehow farthefly hastraveledby thetime thetrainsmeetin themiddleandsquashit.

(d) How long doesit take the trains to meet inthemiddle?Usethis to answerpart(c) withoutsummingaseries.

2. Probability of Winning in SportsIn certainsports,winning a gamerequiresa

leadof two points.That is, if thescoreis tied youhaveto scoretwo pointsin a row to win.

(a) For somesports(e.g.tennis),a point is scoredevery play. Supposeyour probability of scor-ing thenext point is always ò . Then,your op-ponent’sprobabilityof scoringthenext point isalways

� P ò .

(i) What is the probability that you win thenext two points?

(ii) What is theprobability thatyou andyouropponentsplit thenext two points,that is,

thatneitherof you winsbothpoints?

(iii) What is the probability that you split thenext two pointsbut you win the two afterthat?

(iv) Whatis theprobabilitythatyoueitherwinthe next two points or split the next twoandthenwin thenext two afterthat?

(v) Give a formula for your probability � ofwinninga tied game.

(vi) Computeyour probability of winning atied gamewhen ò ) �� - ; when ò ) �� 7 ;when ò ) �� ; when ò ) ��

. Commentonyouranswers.

(vii) In other sports(e.g. volleyball), you canscorea point only if it is your turn, withturns alternatinguntil a point is scored.Supposeyour probability of scoring apoint when it is your turn is ò , andyouropponent’s probability of scoringa pointwhenit is herturn is � .A. Find a formula for the probability a

thatyou arethefirst to scorethenextpoint, assumingit is currently yourturn.

B. Supposethat if you scorea point, thenext turn is yours. Using your an-swersto part(a)andyourformulafora , computethe probability of win-ning a tied game(if you need twopointsin a row to win).Ï Assumeò ) �� - and � ) �� - and

it is your turn.

PROJECTS 455Ï Assumeò ) �� 7 and � ) �� - and it is your turn.

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· Created Date: 1/9/2001 9:45:59 AM