Problems and Solutions in Real Analysis

294 Pages·2007·1.63 MB·English

by  Masayoshi Hata

PROBLEMS AND
SOLUTIONS IN
REAL ANALYSIS
Series on Number Theory and Its Applications ISSN 1793-3161
Series Editor: Shigeru Kanemitsu (Kinki University, Japan)
Editorial Board Members:
V. N. Chubarikov (Moscow State University, Russian Federation)
Christopher Deninger (Universität Münster, Germany)
Chaohua Jia (Chinese Academy of Sciences, PR China)
Jianya Liu (Shangdong University, PR China)
H. Niederreiter (National University of Singapore, Singapore)
M. Waldschmidt (Université Pierre et Marie Curie, France)
Advisory Board:
K. Ramachandra (Tata Institute of Fundamental Research, India (retired))
A. Schinzel (Polish Academy of Sciences, Poland)
Vol. 1 Arithmetic Geometry and Number Theory
edited by Lin Weng & Iku Nakamura
Vol. 2 Number Theory: Sailing on the Sea of Number Theory
edited by S. Kanemitsu & J.-Y. Liu
ZhangJi - Problems and Solutions.pmd 2 8/31/2007, 4:21 PM
Series on Number Theory and Its Applications Vol.4
PROBLEMS AND
SOLUTIONS IN
REAL ANALYSIS
Masayoshi Hata
Kyoto University, Japan
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Number Theory and Its Applications — Vol. 4
PROBLEMS AND SOLUTIONS IN REAL ANALYSIS
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-277-601-3
ISBN-10 981-277-601-X
ISBN-13 978-981-277-949-6 (pbk)
ISBN-10 981-277-949-3 (pbk)
Printed in Singapore.
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July31,2007 13:35 WSPC/BookTrimSizefor9inx6in real-analysis
Preface
Romewasnotbuiltinaday...
There is no shortcut to good scholarship. To learn mathematics you are to
solvemany‘good’problemswithouthaste. Mathematicsisnotonlyforpersons
oftalent. Tacklingdifficultproblemsislikechallengingyourself. Evenifyoudo
not make a success in solving a problem, you may set some new knowledge or
techniqueyoulacked.
This book contains more than one hundred and fifty mathematical problems
andtheirdetailedsolutionsrelatedmainlytoRealAnalysis. Manyproblemsare
selectedcarefullybothforstudentswhoarepresentlylearningorthosewhohave
justfinishedtheircoursesinCalculusandLinearAlgebra,orforanypersonwho
wants to review and improve his or her skill in Real Analysis and, moreover, to
make a step forward, for example, to Complex Analysis, Fourier Analysis, or
Lebesgue Integration, etc. The solutions to all problems are supplied in detail,
which should compete well with the famous books written by Po´lya and Szego¨
morethanthirty-fiveyearsago.
SomeproblemsaretakenfromAnalyticNumberTheory;forexample,theuni-
formdistribution(Chapter12)andtheprimenumbertheorem(Chapter17). The
latteristreatedinaslightlydifferentway. Theymaybeusefulforanintroduction
toAnalyticNumberTheory.
Neverthelessthereadershouldnoticethatallsolutionsarenotshortandele-
gant. Itmayalwaysbepossibleforthereadertofindbetterandmoreelementary
solutions. The problems are merely numbered for convenience’ sake and so the
readershouldgrapplewiththemusinganytools,whichmakesadifferencefrom
theusualexercisesinCalculus. Onemayuseintegrationforproblemsonseries,
forexample. Theauthormustconfessthattherearesomeproblemsexpressedin
an elementary way, whose simple and elementary proof could not be found by
v
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vi ProblemsandSolutionsinRealAnalysis
theauthor. Thereasonwhyhedaredtoincludesuchproblemsandthesolutions
beyondthelimitsofCalculusisleavingtourgethereadertofindbetterones.
The author wishes to take this opportunity to thank Professor S. Kanemitsu
forinvaluablehelpinthepreparationofthemanuscript.
Enjoymathematicswithapen!
M.Hata
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Preface vii
Someremarksonthedefinitionsandnotationsarelistedbelow.
(cid:15) Let f(x)beareal-valuedfunctiondefinedonanopeninterval(a,b)andletc
beanypointin[a,b). Wewrite f(c+)or
lim f(x)
x!c+
todenotetheright-handlimitifitexists.Notethatsomebooksusethenotation
“x!c+0”insteadof“x!c+”. Thisisthelimitof f(x)asxapproachestothe
pointcsatisfyingx>c.Theleft-handlimit f(c(cid:0))canalsobedefinedsimilarly
forc2(a,b]ifitexists.
(cid:15) Theright-handderivativeof f(x)atcisdenotedby f0(x)ifitexists. Wealso
+
definetheleft-handderivative f0(x)ifitexists.
(cid:0)
{ } { }
(cid:15) Giventwosequences a and b withb (cid:21)0foralln(cid:21)1,wewrite
n n n
a =O(b )
n n
ifja j (cid:20) Cb holdsforalln (cid:21) 1withsomepositiveconstantC. Inparticular,
n n { }
a = O(1)isnothingbuttheboundednessofthesequence a . Thisnotation
n n
ofLandauisaconvenientwayofexpressinginequalities. Notethatthesymbol
O(b ) is not a specific sequence; thus one can write O(1) + O(1) = O(1),
n
forexample. Usuallyweusethesenotationstoshowtheasymptoticbehavior
{ } { }
of a . Themajorantsequence b maybechosenamongstandardpositive
n n
sequencessuchasnα,(logn)β,eδn,etc.,where
( )
e= lim 1+ 1 n = ∑1 1
n!1 n n!
n=0
isthebasetothenaturallogarithm.
(cid:15) Ifb > 0forallsufficientlylargenanda /b convergesto0asn ! 1,then
n n n
wewrite
a =o(b ).
n n
Inparticular,a =o(1)meanssimplythata convergesto0asn!1.
n n
(cid:15) Ifthelimitofa /b existsandequalsto1,wewrite
n n
a (cid:24)b
n n
{ }
whichgivesclearlyanequivalencerelation. Inthiscasewesaythat a and
{ } { } { } {n }
b are asymptotically equivalent or that a is asymptotic to b . b is
n n n n
July31,2007 13:35 WSPC/BookTrimSizefor9inx6in real-analysis
viii ProblemsandSolutionsinRealAnalysis
{ }
alsosaidtobetheprincipalpartof a . Notethata (cid:24)b ifandonlyif
n n n
a =b +o(b ).
n n n
(cid:15) Landau’snotationscanalsobeappliedtoexpresstheasymptoticbehaviorofa
givenfunction f(x)as x ! c. Ifjf(x)j (cid:20) Cg(x)holdsonasufficientlysmall
neighborhoodofthepointcforsomenon-negativefunctiong(x)andpositive
constantC,wewrite
f(x)=O(g(x))
as x ! c. We can also define f(x) = o(g(x)) and f(x) (cid:24) g(x) in the same
manneraswithsequences,evenwhenx!1orx!(cid:0)1.
(cid:15) The set of all continuous functions f(x) defined on an interval I possessing
thecontinuousntimesderivative f(n)(x)isdenotedbyCn(I). IftheintervalI
containsanendpoint,thederivativeatthispointmayberegardedastheone-
sidedderivative. Inparticular,thesetofallcontinuousfunctionsdefinedon I
isdenotedbyC(I).
(cid:15) Thesignfunctionsgn(x)isdefinedby

 1 for x>0,
sgn(x)= 0 for x=0,
(cid:0)1 for x<0.
(cid:15) Wewillreasonablysuppressorabbreviateparenthesesusedinsomecases. For
example,wewritesinnθandsin2xinsteadofsin(nθ)and(sinx)2respectively.
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Contents
Preface v
1. SequencesandLimits 1
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. InfiniteSeries 15
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. ContinuousFunctions 31
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4. Differentiation 43
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5. Integration 59
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6. ImproperIntegrals 77
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7. SeriesofFunctions 93
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8. ApproximationbyPolynomials 113
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
ix
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x ProblemsandSolutionsinRealAnalysis
9. ConvexFunctions 125
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10. Variousproofsofζ(2)=π2/6 139
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
11. FunctionsofSeveralVariables 157
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
12. UniformDistribution 171
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
13. RademacherFunctions 181
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14. LegendrePolynomials 191
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
15. ChebyshevPolynomials 205
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
16. GammaFunction 219
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
17. PrimeNumberTheorem 239
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
18. Miscellanies 257
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Bibliography 273
Index 285