A First Course in Mathematical Analysis

470 Pages·2006·3.762 MB·English

by  David Alexander Brannan

A First Course in Mathematical Analysis
MathematicalAnalysis(oftencalledAdvancedCalculus)isgenerallyfoundbystudents
tobeoneoftheirhardestcoursesinMathematics.Thistextusestheso-calledsequential
approachtocontinuity,differentiabilityandintegrationtomakeiteasiertounderstand
thesubject.
Topics that are generally glossed over in the standard Calculus courses are given
careful study here. For example, what exactly is a ‘continuous’ function? And how
exactlycanonegiveacareful definitionof‘integral’?Thislatterisoftenoneofthe
mysterious points in a Calculus course – and it is quite tricky to give a rigorous
treatmentofintegration!
Thetexthasalargenumberofdiagramsandhelpfulmarginnotes,andusesmany
graded examples and exercises, often with complete solutions, to guide students
throughthetrickypoints.Itissuitableforselfstudyoruseinparallelwithastandard
universitycourseonthesubject.
A First Course in
Mathematical
Analysis
D A V I D AL E X A N D ER B R A N N A N
PublishedinassociationwithTheOpenUniversity
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521864398
© The Open University 2006
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2006
ISBN-13 978-0-511-34857-0 eBook (EBL)
ISBN-10 0-511-34857-6 eBook (EBL)
ISBN-13 978-0-521-86439-8 hardback
ISBN-10 0-521-86439-9 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
TomywifeMargaret
andmysonsDavid,JosephandMichael
Contents
Preface pageix
1 Numbers 1
1.1 Realnumbers 2
1.2 Inequalities 9
1.3 Provinginequalities 14
1.4 Leastupperboundsandgreatestlowerbounds 22
1.5 Manipulatingrealnumbers 30
1.6 Exercises 35
2 Sequences 37
2.1 Introducingsequences 38
2.2 Nullsequences 43
2.3 Convergentsequences 52
2.4 Divergentsequences 61
2.5 TheMonotoneConvergenceTheorem 68
2.6 Exercises 79
3 Series 83
3.1 Introducingseries 84
3.2 Serieswithnon-negativeterms 92
3.3 Serieswithpositiveandnegativeterms 103
3.4 Theexponentialfunctionx7!ex 122
3.5 Exercises 127
4 Continuity 130
4.1 Continuousfunctions 131
4.2 Propertiesofcontinuousfunctions 143
4.3 Inversefunctions 151
4.4 Definingexponentialfunctions 161
4.5 Exercises 164
5 Limitsandcontinuity 167
5.1 Limitsoffunctions 168
5.2 Asymptoticbehaviouroffunctions 176
5.3 Limitsoffunctions–using"and(cid:2) 181
5.4 Continuity–using"and(cid:2) 193
5.5 Uniformcontinuity 200
5.6 Exercises 203
vii
viii Contents
6 Differentiation 205
6.1 Differentiablefunctions 206
6.2 Rulesfordifferentiation 216
6.3 Rolle’sTheorem 228
6.4 TheMeanValueTheorem 232
6.5 L’Hoˆpital’sRule 238
6.6 TheBlancmangefunction 244
6.7 Exercises 252
7 Integration 255
7.1 TheRiemannintegral 256
7.2 Propertiesofintegrals 272
7.3 FundamentalTheoremofCalculus 282
7.4 Inequalitiesforintegralsandtheirapplications 288
7.5 Stirling’sFormulaforn! 303
7.6 Exercises 309
8 Powerseries 313
8.1 Taylorpolynomials 314
8.2 Taylor’sTheorem 320
8.3 Convergenceofpowerseries 329
8.4 Manipulatingpowerseries 338
8.5 Numericalestimatesforp 346
8.6 Exercises 350
Appendix1 Sets,functionsandproofs 354
Appendix2 Standardderivativesandprimitives 359
p
ffiffiffi
Appendix3 Thefirst1000decimalplacesof 2,eandp 361
Appendix4 Solutionstotheproblems 363
Chapter1 363
Chapter2 371
Chapter3 382
Chapter4 393
Chapter5 402
Chapter6 413
Chapter7 426
Chapter8 443
Index 457
Preface
Analysis is a central topic in Mathematics, many of whose branches use
key analytic tools. Analysis also has important applications in Applied
Mathematics, Physics and Engineering, where a good appreciation of the
underlyingideasofAnalysisisnecessaryforamoderngraduate.
Changes in the school curriculum over the last few decades have resulted
in many students finding Analysis very difficult. The author believes that
Analysis nowadays has an unjustified reputation for being hard, caused by
thetraditionaluniversityapproachofprovidingstudentswithahighlypolished
exposition in lectures and associated textbooks that make it impossible for
theaveragelearnertograspthecoreideas.Manystudentsendupagreeingwith
theGermanpoetandphilosopherGoethewhowrotethat‘Mathematiciansare JohannWolfgangvonGoethe
like Frenchmen: whatever you say to them, they translate into their own (1749–1832)issaidtohave
studiedallareasofscienceof
language,andforthwithitissomethingentirelydifferent!’
hisdayexceptmathematics–
Since1971,theOpenUniversityinUnitedKingdomhastaughtMathematics
forwhichhehadnoaptitude.
tostudentsintheirownhomesviaspeciallywrittencorrespondencetexts,and
hastraditionallygivenAnalysisacentralpositioninitscurriculum.Itsphilo-
sophyistoprovideclearandcompleteexplanationsoftopics,andtoteachthese
inawaythatstudentscanunderstandwithoutmuchexternalhelp.Asaresult,
studentsshouldbeabletolearn,andtoenjoylearning,thekeyconceptsofthe
subjectinanunclutteredway.Thisbookarisesfromcorrespondencetextsfor
its course Introduction to Pure Mathematics, that has now been studied suc-
cessfullybyovertenthousandstudents.
ThisbookisthereforedifferentfrommostMathematicstextbooks!Itadopts
astudent-friendlyapproach,beingdesignedforstudybyastudentontheirown
ORinparallelwithacoursethatusesassettexteitherthistextoranothertext.
Butthisisthetextthatthestudentislikelytousetolearnthesubjectfrom.The
author hopes that readers will gain enormous pleasure from the subject’s
beauty and that this will encourage them to undertake further study of
Mathematics!
Once a student has grasped the principal notions of limit and continuous
functionintermsofinequalitiesinvolvingthethreesymbols",Xand(cid:2),they
willquicklyunderstandtheunityofareasofAnalysissuchaslimits,continu-
ity, differentiability and integrability. Then they will thoroughly enjoy the
beautyofsomeoftheargumentsusedtoprovekeytheorems–whethertheir
proofsareshortorlong.
Calculusistheinitialstudyoflimits,continuity,differentiationandintegra-
tion, where functions are assumed to be well-behaved. Thus all functions
continuous on an interval are assumed to be differentiable at most points in
theinterval,andsoon.However,Mathematicsisnotthatsimple!Forexample,
there existfunctions that are continuous everywhere on R, but differentiable
ix
x Preface
nowhereonR;thisdiscoverybyKarlWeierstrassin1872causedasensationin
the mathematical community. In Analysis (sometimes called Advanced
Calculus) we make no assumptions about the behaviour of functions – and
theresultisthatwesometimescomeacrossrealsurprises!
The book has two principal features in its approach that make it stand out
fromamongotherAnalysistexts.
Firstly, this book uses the ‘sequential approach’ toAnalysis. Alltoo often
studentsstartingonthesubjectfindthattheycannotgraspthesignificanceof
both"and(cid:2)simultaneously.Thismeansthatthewholeunderlyingideaabout
whatishappeningislost,andthestudenttakesaverylongtimetomasterthe
topic–or,inmanycasesinfact,nevermastersthetopicandacquiresastrong
dislikeofit.Inthesequentialapproachtheyproceedatamoreleisurelypaceto
understandthenotionoflimitusing"andX–tohandleconvergentsequences–
beforecomingacrosstheothersymbol(cid:2),usedinconjunctionwith"tohandle
continuousfunctions.Thisapproachavoidstheconventionalstudenthorrorat
theperceived‘difficultyofAnalysis’.Also,itavoidsthenecessitytore-prove
broadlysimilarresultsinarangeofsettings–forexample,resultsonthesum
of two sequences, of two series, of two continuous functions and of two
differentiablefunctions.
Secondly,thisbookmakesgreateffortstoteachthe"–(cid:2)approachtoo.After
students have had a first pass at convergence of sequences and series and at
continuityusing‘thesequentialapproach’,theythenmeet‘the"–(cid:2)approach’,
explained carefully and motivated by a clear ‘"–(cid:2) game’ discussion. This
makes the new approach seem very natural, and this is motivated by using
eachapproachinlaterworkintheappropriatesituation.Bytheendofthebook,
studentsshouldhaveagoodfacilityatusingboththesequentialapproachand
the"–(cid:2)approachtoproofsinAnalysis,andshouldbebetterpreparedforlater
studyofAnalysisthanstudentswhohaveacquiredonlyaweakunderstanding
oftheconventionalapproach.
Outline of the content of the book
InChapter1,wedefinerealnumberstobedecimals.Ratherthangiveaheavy
discussion of least upper bound and greatest lower bound, we give an intro-
ductiontothesematterssufficientforourpurposes,andthefulldiscussionis
postponed to Chapter 7, where it is more timely. We also study inequalities,
andtheirpropertiesandproofs.
In Chapter 2, we define convergent sequences and examine their proper-
ties, basing the discussion on the notion of null sequence, which simplifies
matters considerably. We also look at divergent sequences, sequences
defined by recurrence formulas and particular sequences which converge
topande.
InChapter3,wedefineconvergentinfiniteseries,andestablishanumberof
tests for determining whether a given series is convergent or divergent. We
demonstratetheequivalenceofthetwodefinitionsoftheexponentialfunction
x7!ex,andprovethatthenumbereisirrational. Wedefineexas lim(cid:2)1þx(cid:3)n
InChapter4,wedefinecarefullywhatwemeanbyacontinuousfunction,in andas P1 xn. n!1 n
termsofsequences,andestablishthekeypropertiesofcontinuousfunctions. n!
n¼0
Wealsogivearigorousdefinitionoftheexponentialfunctionx7!ax.
Preface xi
InChapter5,wedefinethelimitofafunctionasxtendstocorasxtends
to 1 in terms of the convergence of sequences. Then we introduce the "–(cid:2)
definitions oflimit andcontinuity,andcheckthattheseareequivalenttothe
earlier definitions in terms of sequences. We also look briefly at uniform
continuity.
In Chapter 6, we define what we mean by a differentiable function, using
differencequotientsQ(h);thisenablesustouseourearlierresultsonlimitsto
prove corresponding results for differentiable functions. We establish some
interesting properties of differentiable functions. Finally, we construct the
Blancmange function that is continuous everywhere on R, but differentiable
nowhereonR.
InChapter7,wegiveacarefuldefinitionofwhatwemeanbyanintegrable
function, and establish a number of related criteria for establishing whether
a given function is integrable or not. Our integral is the so-called Riemann
integral, defined in terms of upper and lower Riemann sums. We check the
standardpropertiesofintegralsandverifyanumberofstandardapproachesfor
calculating definite integrals. Then we give a number of applications of
integralstolimitsofcertainsequencesandseriesandproveStirling’sFormula. Stirling’sFormulasaysthat,
Finally, in Chapter 8, we study the convergence and properties of power fporlargen,n!is‘roughly’
ffi2ffiffipffiffiffinffiffi(cid:2)n(cid:3)n,inasensethatwe
series. The chapter ends with a marvellous proof of the irrationality of the e
explain.
numberpthatusesawholerangeofthetechniquesthathavebeenmetinthe
previouschapters.
Forcompletenessandforstudents’convenience,wegiveabriefguidetoour
notation for sets and functions, together with a brief indication of the logic
involvedinproofsinMathematics(inparticular,thePrincipleofMathematical
Induction) in Appendix 1. Appendix 2 contains a list of standard derivatives
andprimitivesandAppendix3thefirst1000decimalplacesinthevaluesofthe
p
ffiffiffi
numbers 2,pande.Appendix4containsfullsolutionstoalltheproblemsset
duringeachchapter.
Solutionsarenotgiventotheexercisesattheendofeachchapter,however.
Lecturers/instructorsmaywishtousetheseexercisesinhomeworkassignments.
Study guide
This book assumes that students have a fair understanding of Calculus. The
assumptionsontechnicalbackgroundaredeliberatelykeptslight,however,so
thatstudentscanconcentrateontheneweraspectsofthesubject‘Analysis’.
Most students will have met some of the material in the early chapters
previously.Althoughthismeansthattheycanthereforeproceedfairlyquickly
throughsomesections,itdoesNOTmeanthatthosesectionscanbeignored–
eachsectioncontainsimportantideasthatareusedlateronandmostinclude
somethingneworhaveadifferentemphasis.
Mostchaptersaredividedintofiveorsixsections(eachoftenfurtherdivided
into sub-sections); sections are numbered using two digits (such as ‘Section
3.2 ’) and sub-sect ions using thr ee digits (such as ‘Sub- section 3.2.4 ’).
Generally a section is considered to be about one evening’s work for an
averagestudent.
Chapter 7 on Integration is arguably the highlight of the book. However,
it contains some rather complicated mathematical arguments and proofs.