Ludmila Bourchtein Andrei Bourchtein Theory of Infinite Sequences and Series Ludmila Bourchtein • Andrei Bourchtein Theory of Infinite Sequences and Series LudmilaBourchtein AndreiBourchtein InstituteofPhysicsandMathematics InstituteofPhysicsandMathematics FederalUniversityofPelotas FederalUniversityofPelotas Pelotas,Brazil Pelotas,Brazil ISBN978-3-030-79430-9 ISBN978-3-030-79431-6 (eBook) https://doi.org/10.1007/978-3-030-79431-6 MathematicsSubjectClassification:40-01,40A05,26-01 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG 2022 Thisworkissubjecttocopyright. 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Neitherthepublishernortheauthorsortheeditors giveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissions thatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmaps andinstitutionalaffiliations. ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.com,bytheregisteredcompany SpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To Victoriaforstimulatingmathematical discussions To Valentinaforexcellence in mathematicaltests To Maximand Nataliaforverydeep learningand dedication To Haimand Mariafor everything Preface This book covers a major part of the theory of real infinite sequences and series at the level corresponding to advanced undergraduate/initialgraduate courses of mathematical analysis. This theory is a natural part of real analysis, but its place in the major subject is ambiguous. On the one hand, the entire course of the real analysis can be based on the infinite sequences and series, but on the other hand, there are expositions of the real analysis topics with no explicit reference to the sequences and series. Some textbooks consider sequences and series as the first topic to be studied, while others leave it for more advanced parts of the text. The level of exposition also varies significantly. The calculus textbooks usually contain the basic exposition of the theory of sequences and series, buttheyare moreconcernedwith the well-knowntests forseries ofnumbersand representationsof functionsin power series, which provide useful tools for the solution of different problems. However, the logical development of the material and rigorous proofsofthepresentedresultsareleftasideduetoobviouslimitsoftimeandthereader’s expertise.Manyrealanalysistextscontainalogicalanddeepexpositionofprincipalparts ofthetheoryofsequencesandseries,buttheynaturallyadjusttherepresentationofthese topics for the main purpose of the book, which usually leads to the separation of the continuoustheoryintothepartsrelatedtosequencesofnumbers,seriesofnumbers,then tosequencesandseriesoffunctions,andfinallytothepowerseries.Thus,duetodifferent reasons,calculusandanalysisbooksusuallyfailtoprovidethepresentationofthetheory ofsequencesandserieswiththelogicalcontinuityandintegrityinherentinthistheory.At thesametime,thereareafewbooksdedicatedespeciallytothesequencesandseries,and eachoneofthemcoversapartofthesubjectduetoawiderangeofitstopics. Our intention is to provide a text that covers the majority of traditional topics of infinite sequences and series and represents the theory of this subject in its integrity, logicalsequence,andsufficientdepth.Itstartsfromtheverybeginning—thedefinitionand elementarypropertiesofsequencesofnumbers,andendswithadvancedresultsofuniform convergenceand powerseries. The entire textis developedat two levels: the basic level covers the undergraduatetopics, while the additional material (marked as Complement) addresses more advanced subjects. The reader can choose what level of the exposition vii viii Preface to follow, since the explanation of more complex problems does not interfere with the developmentandcomprehensionofthetopicsofthebasiclevel. This book is aimed at university students specializing in mathematics and natural sciences, and at readersinterested in infinite sequencesand series. It is designed for the readerwhohasagoodworkingknowledgeofcalculus.Noadditionalpriorknowledgeis required.Alltheinitialconceptsandresultsrelatedtosequencesandseriesandrequired for the development of the theory are covered in the initial part of the text and, when necessary,atthestartingpointofeachtopic.Thismakesthebookself-sufficientandthe readerindependentofanyothertextonthissubject.Asa consequence,thisbookcanbe usedbothasatextbookforadvancedundergraduate/earlygraduatecoursesandasasource forself-studyofselected(orall)topicspresentedinthetext. Thetextisdividedintofivechapterswhichcanbegroupedintotwoparts:thefirsttwo chapters are concerned with the sequences and series of numbers, while the remaining three chaptersare devotedto the sequencesand series of functions,includingthe power series. Eachchapteriscomprisedofanumberofsectionsandsubsections,whicharenumbered separatelywithineachchapter,bearingthesequentialnumberofsectioninsideachapter and that of subsection inside a section. For instance, Sect.3 can be found in each of the five chapters, while Sect.3.2 means the second subsection inside the third section. When referring to a section or subsection inside the current chapter, we do not use the chapter number, otherwise the chapter number is provided. The formulas and figures are numbered sequentially within each chapter regardless of the section where they are found.Inthisway,formula(2.5)isthereferencetothefifthformulainthesecondchapter, whileFig.3.4indicatesthefourthfigureinthethirdchapter.Thetheorems,propositions, lemmas,definitions,andexamplesarenumberedindependentlywithineachsection. Throughoutthetext,themostimportant,fundamental,andusefuldefinitionsandresults are highlighted, but this reflects only the personal opinion of the authors and can be disputed.Briefhistoricalcommentsonsomesignificantresultsandinterestingfactsonthe developmentof the theory of sequencesand series are providedat the end of individual sections. Within each major topic, the exposition is, as a rule, inductive and starts with rather simpledefinitionsand/orexamples,becomingmorecompressedandsophisticatedasthe courseprogresses.Eachimportantnotionandresultisillustratedwithexamplesexplained indetail.Somemorecomplicatedtopicsandresultsaremarkedascomplementsandcan beomittedonafirstreadingwithoutlossofsubjectcontinuity.Themathematicallevelof the exposition correspondsto advancedundergraduatecourses of mathematical analysis and/orearlygraduateintroductiontothediscipline. The first chapter introduces the initial concepts of sequences of numbers and their convergence. It studies different properties of convergent sequences, first similar to the limitpropertiesofgeneralfunctionsandthenspecificpropertiesofsequences.Attheend of this chapter, we explain the methods of solution of indeterminate forms, frequently usedinsubsequentchapters.Basedonthedevelopedmaterialofthesequences,thesecond Preface ix chapterdealswiththeseriesofnumbers.Wepresentastandardsetoftheconvergencetests thatcanbefoundincalculusandanalysistextbooks(integral,comparison,ratio,androot tests for positiveseries, andLeibniz’stest foralternatingseries) in primitiveand refined forms. We also investigatethe two principal chainsof more elaboratedtests for positive series—the Kummer and Cauchy chains, and more strong tests for arbitrary series— Dirichlet’s and Abel’s tests. At the end of this chapter, we analyze the associative and commutativepropertiesofseries,includingthefamousDirichletandRiemanntheorems. The part of the text devoted to the study of sequences and series of functions starts with the third chapter. The principal theme of this chapter is the uniform convergence of sequences, the methods of its investigation, and the conditions which guarantee “nice” properties of limit functions, such as boundedness, continuity, integration, and differentiationby parameter. This sets the stage for developingthe material on series of functionscoveredinthefourthchapter.Theuniformconvergencecontinuestobethefocus of the study,and the expositionrevolveson the one handaroundthe conditions(criteria andtests)thatensuretheuniformconvergence,andontheotherhandaroundhypotheses basedontheuniformconvergencethatprovidethedesiredpropertiesofthesumsofseries. The fifth chapter deals with the theory of power series, which, as a particular case of the series of functions, inherits many properties of the latter, but also has its specific characteristics. Different techniques of finding power series expansions are considered andemployedtoderivetherepresentationofmanyelementaryfunctions.Attheendofthe chapter,thestandardapplicationsofpowerseriesareconsidered,includingtheproblems of approximation, calculation of limits, integration, and solving ordinary differential equations. The text contains a large number of problems and exercises, which should make it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test understanding of concepts. Other problemsaremoretheoreticallyorientedandillustratemoreintricatepointsofthetheory orprovidecounterexamplestofalsepropositionswhichseemtobenaturalatfirstglance. Someharderexercisesoftheoreticalinterestarealsoincludedasexamplesorapplications of theoretical results, but they may be omitted in courses addressed to less advanced students. Many additionalproblems are proposed as homework tasks at the end of each chapter. Their level ranges from straightforward, but not overly simple, exercises to problemsofconsiderabledifficulty,butofcomparableinterest.Thesemoreinvolvedand challengingproblemsaremarkedwithanasterisk. Thepresentedtexthasthefollowingfeatures: 1. Completeness:the text coversa major part of the traditionaltopics of real sequences andseriesattheadvancedundergraduate/initialgraduatelevel. 2. Self-sufficiency:allthebackgroundtopicsrelatedtosequencesandseriesarecovered inthetext,and,consequently,thisworkcanbeusedasbothatextbookandasourcefor self-study. x Preface 3. Generality:wehaveendeavoredtopresentalltheresultsinamoregeneralformwhile avoidingmajorcomplicationsoftheirproofs. 4. Accessibility: all the topics are covered in a rigorous mathematical manner while keepingtheexpositionatalevelacceptableforadvancedundergraduatecourses. 5. Two-levelapproach:thetextissystematically developedattwo levels—thebasic and moreadvanced,withthepossibilitytochoosewhatlevelofexpositiontofollow. 6. Exercises: there are a large number of problemsand exercises, solved and proposed, whichshouldmakethebooksuitableforbothclassroomuseandself-study. Pelotas,Brazil LudmilaBourchtein AndreiBourchtein Contents 1 SequencesofNumbers.............................................................. 1 1 ConvergenceandIntroductoryExamples....................................... 1 1.1 DefinitionofaSequenceandTrivial(Pre-limit)Properties............ 1 1.2 ConvergenceofaSequence.............................................. 3 2 CommonPropertiesofConvergentSequences................................. 8 2.1 UniquenessoftheLimit.................................................. 8 2.2 ComparisonProperties................................................... 8 2.3 ArithmeticandAnalyticProperties...................................... 11 3 SpecialPropertiesofConvergentSequences ................................... 13 3.1 ConvergenceofFunctionandCorrespondingSequence ............... 13 3.2 RelationshipBetweenConvergenceandBoundedness................. 13 3.3 SubsequencesandTheirConvergence.Bolzano-Weierstrass Theorem................................................................... 14 3.4 CauchyCriterionforConvergence ...................................... 18 3.5 SequencesoftheArithmeticandGeometricMeans.................... 19 4 IndeterminateFormsandTechniquesofTheirSolution ....................... 22 4.1 DefinitionofIndeterminateForms....................................... 22 4.2 TechniquesofSolutionofIndeterminateForms........................ 24 4.3 VariousIndeterminateFormsandExamples............................ 36 Exercises.............................................................................. 38 2 SeriesofNumbers................................................................... 43 1 ConvergenceandIntroductoryExamples....................................... 43 1.1 DefinitionofaSeries.PartialSumsandConvergence.................. 43 1.2 ElementaryExamplesofSeriesofNumbers............................ 46 2 ElementaryPropertiesofConvergentSeries.................................... 52 2.1 ArithmeticProperties..................................................... 53 2.2 CauchyCriterionforConvergence ...................................... 54 2.3 NecessaryConditionofConvergence(DivergenceTest)............... 55 2.4 SeriesandItsRemainder................................................. 55 xi