Undergraduate Texts in Mathematics Charles H.C. Little Kee L. Teo Bruce van Brunt Real Analysis via Sequences and Series Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege DavidA.Cox,AmherstCollege PamelaGorkin,BucknellUniversity RogerE.Howe,YaleUniversity MichaelOrrison,HarveyMuddCollege JillPipher,BrownUniversity FadilSantosa,UniversityofMinnesota Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Moreinformationaboutthisseriesathttp://www.springer.com/series/666 Charles H.C. Little • Kee L. Teo (cid:129) Bruce van Brunt Real Analysis via Sequences and Series 123 CharlesH.C.Little KeeL.Teo ResearchFellowandformer ResearchFellowandformer ProfessorofMathematics AssociateProfessorofMathenatics InstituteofFundamentalSciences InstituteofFundamentalSciences MasseyUniversity MasseyUniversity PalmerstonNorth,NewZealand PalmerstonNorth,NewZealand BrucevanBrunt AssociateProfessorofMathematics InstituteofFundamentalSciences MasseyUniversity PalmerstonNorth,NewZealand ISSN0172-6056 ISSN2197-5604 (electronic) UndergraduateTextsinMathematics ISBN978-1-4939-2650-3 ISBN978-1-4939-2651-0 (eBook) DOI10.1007/978-1-4939-2651-0 LibraryofCongressControlNumber:2015935731 MathematicsSubjectClassification(2010):26-01 SpringerNewYorkHeidelbergDordrechtLondon ©SpringerScience+BusinessMediaNewYork2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerScience+BusinessMediaLLCNewYorkispartofSpringerScience+BusinessMedia(www. springer.com) Preface Thisbookisatextonrealanalysisforstudentswithabasicknowledgeofcalculus of a single variable. There are many fine works on analysis, and one must ask what advantages a new book brings. This one contains the standard material for a first course in analysis, but our treatment differs from many other accounts in thatconceptssuchascontinuity,differentiation,andintegrationareapproachedvia sequences.Themainanalyticalconceptisthustheconvergenceofasequence,and thisideaisextendedtodefineinfiniteseriesandlimitsoffunctions.Thisapproach not only has the merit of simplicity but also places the student in a position to appreciate and understand more sophisticated concepts such as completeness that playacentralpartinmoreadvancedfieldssuchasfunctionalanalysis. The theory of sequences and series forms the backbone of this book. Much of the material in the book is devoted to this theory and, in contrast to many other texts, infinite series are treated early. The appearance of series in Chap. 3 has the advantages that it provides many straightforward applications of the results for sequences given in Chap. 2 and permits the introduction of the elementary tran- scendentalfunctionsasinfiniteseries.Thedisadvantageisthatcertainconvergence testssuchastheintegraltestmustbepostponeduntiltheimproperintegralisdefined inChap.7.TheCauchycondensationtestisusedinChap.3totackleconvergence problemswheretheintegraltestisnormallyapplied.Althoughmuchofthematerial in Chap. 2 is standard, there are some unusual features such as the treatment of harmonic, geometric, and arithmetic means and the sequential definition of the exponential function. In Chap. 3 we present results, such as the Kummer–Jensen test,Dirichlet’stest,andRiemann’stheoremontherearrangementofseries,thatare oftenpostponedornottreatedinafirstcourseinanalysis. Limits of functions are introduced in Chap. 4 through the use of convergent sequences,andthisconceptisthenusedinChaps.5and6tointroducecontinuity anddifferentiation.Aswithanyanalysisbook,resultssuchastheintermediate-value theoremandthemean-valuetheoremcanbefoundinthesechapters,butthereare also some other features. For instance, the logarithm is introduced in Chap. 5 and thenusedtoproveGauss’stestforinfiniteseries.InChap.6wepresentadiscrete v vi Preface versionofl’Hôpital’srulethatisseldomfoundinanalysistexts.Weconcludethis chapterwithashortaccountofthedifferentiationofpowerseriesusingdifferential equationstomotivatethediscussion. The Riemann integral is presented in Chap. 7. In the framework of elementary analysis this integral is perhaps more accessible than, say, the Lebesgue integral, anditisstillanimportantconcept.Thischapterfeaturesanumberofitemsbeyond the normal fare. In particular, a proof of Wallis’s formula followed by Stirling’s formula, and a proof that (cid:2) and e are irrational, appear here. In addition, there is alsoashortsectiononnumericalintegrationthatfurtherillustratesthedefinitionof theRiemannintegralandapplicationsofresultssuchasthemean-valuetheorem. Chapter 8 consists of a short account of Taylor series. Much of the analytical apparatusforthistopicisestablishedearlierinthebooksothat,asidefromTaylor’s theorem, the chapter really covers mostly the mechanics of determining Taylor series.Thereisanextensivetheoryonthistopic,anditisdifficulttolimitoneselfso severelytothesebasicideas.Here,despitethebook’semphasisonseries,theauthors eschewtopicssuchasthetheoremsofAbelandTauberand,moreimportantly,the questionofwhichfunctionshaveaTaylorseries.Afullappreciationofthistheory requirescomplexanalysis,whichtakesustoofarafield. The student encounters Newton’s method in a first calculus course as an application of differentiation. This method is based on constructing a sequence, motivated geometrically, that converges (hopefully) to the solution of a given equation. The emphasis in this first encounter is on the mechanics of the method and choosing a sensible “initial guess.” In Chap. 9 we look at this method in the widercontextofthefixed-pointproblem.Fixed-pointproblemsprovideapractical applicationofthetheoryofsequences.ThesequenceproducedbyNewton’smethod isalreadyfamiliartothestudent,andthetheoryshowshowproblemssuchaserror estimatesandconvergencecanberesolved. Thefinalchapterdealswithsequencesoffunctionsanduniformconvergence.By this stage the reader is familiar with the example of power series, but those series have particularly nice properties not shared generally by other series of functions. Thematerialismotivatedbyaproblemindifferentialequationsfollowedbyvarious examples that illustrate the need for more structure. This chapter forms a short introductiontothefieldandismeanttoprimethereaderformoreadvancedtopics inanalysis. Anintroductorycourseinanalysisisoftenthefirsttimeastudentisexposedto the rigor of mathematics. Upon reflection, some students might even view such a course as a rite of passage into mathematics, for it is here that they are taught the need for proofs, careful language, and precise arguments. There are few shortcuts to mastering the subject, but there are certain things a book can do to mitigate difficultiesandkeepthestudentinterestedinthematerial.Inthisbookwestriveto motivatedefinitions,results,andproofsandpresentexamplesthatillustratethenew material. These examples are generally the simplest available that fully illuminate the material. Where possible, we also provide examples that show why certain conditions are needed. A simple counterexample is an exceedingly valuable tool Preface vii forunderstandingandrememberingaresultthatisladenwithtechnicalconditions. There are exercises at the end of most sections. Needless to say, it is here that the studentbeginstofullyunderstandthematerial. Theauthorsappreciatetheencouragementandsupportoftheirwives.Theyalso thank Fiona Richmond for her help in preparing the figures. The work has also benefitedfromthethoughtfulcommentsandsuggestionsofthereviewers. PalmerstonNorth,NewZealand CharlesH.C.Little KeeL.Teo BrucevanBrunt Contents 1 Introduction................................................................. 1 1.1 Sets.................................................................... 1 1.2 OrderedPairs,Relations,andFunctions ............................ 3 1.3 InductionandInequalities ........................................... 5 1.4 ComplexNumbers ................................................... 14 1.5 FiniteSums........................................................... 23 2 Sequences.................................................................... 33 2.1 DefinitionsandExamples............................................ 33 2.2 ConvergenceofSequences........................................... 37 2.3 AlgebraofLimits..................................................... 43 2.4 Subsequences......................................................... 53 2.5 TheSandwichTheorem.............................................. 55 2.6 TheCauchyPrinciple ................................................ 65 2.7 MonotonicSequences................................................ 77 2.8 UnboundedSequences............................................... 104 3 Series......................................................................... 109 3.1 Introduction........................................................... 109 3.2 DefinitionofaSeries................................................. 110 3.3 ElementaryPropertiesofSeries ..................................... 117 3.4 TheComparisonTest................................................. 121 3.5 Cauchy’sCondensationTest......................................... 123 3.6 TheLimitComparisonTest.......................................... 127 3.7 TheRatioTest........................................................ 132 3.8 TheRootTest......................................................... 136 3.9 TheKummer–JensenTest............................................ 141 3.10 AlternatingSeries .................................................... 147 3.11 Dirichlet’sTest ....................................................... 152 3.12 AbsoluteandConditionalConvergence............................. 156 3.13 RearrangementsofSeries............................................ 160 3.14 ProductsofSeries .................................................... 165 ix