Problems in Real Analysis Advanced Calculus on The Real Axis

Advanced Calculus on the Real Axis

Teodora-Liliana T. R˘adulescu Vicen¸tiu D. R˘adulescu Department of Mathematics

Simion Stoilow Mathematics Institute Fratii Buzesti National College

Romanian Academy Craiova 200352

Bucharest 014700 Romania

Romania

teodoraradulescu@yahoo.com vicentiu.radulescu@math.cnrs.fr

Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas Richardson, TX 75080 USA titu.andreescu@utdallas.edu

ISBN: 978-0-387-77378-0

e-ISBN: 978-0-387-77379-7

DOI: 10.1007/978-0-387-77379-7 Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2009926486 Mathematics Subject Classification (2000): 00A07, 26-01, 28-01, 40-01 © Springer Science+Business Media, LLC 2009

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To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. —George P´olya (1887–1985)

We come nearest to the great when we are great in humility. —Rabindranath Tagore (1861–1941)

Foreword

This carefully written book presents an extremely motivating and original approach, by means of problem-solving, to calculus on the real line, and as such, serves as a perfect introduction to real analysis. To achieve their goal, the authors have care- fully selected problems that cover an impressive range of topics, all at the core of the subject. Some problems are genuinely difficult, but solving them will be highly rewarding, since each problem opens a new vista in the understanding of mathematics. This book is also perfect for self-study, since solutions are provided.

I like the care with which the authors intersperse their text with careful reviews of the background material needed in each chapter, thought-provoking quotations, and highly interesting and well-documented historical notes. In short, this book also makes very pleasant reading, and I am confident that each of its readers will enjoy reading it as much as I did. The charm and never-ending beauty of mathematics pervade all its pages.

In addition, this little gem illustrates the idea that one cannot learn mathematics without solving difficult problems. It is a world apart from the “computer addiction” that we are unfortunately witnessing among the younger generations of would-be mathematicians, who use too much ready-made software instead or their brains, or who stand in awe in front of computer-generated images, as if they had become the essence of mathematics. As such, it carries a very useful message.

One cannot help comparing this book to a “great ancestor,” the famed Problems and Theorems in Analysis , by P´olya and Szeg˝o, a text that has strongly influenced generations of analysts. I am confident that this book will have a similar impact.

Hong Kong, July 2008 Philippe G. Ciarlet

Preface

If I have seen further it is by standing on the shoulders of giants. —Sir Isaac Newton (1642–1727), Letter to Robert Hooke, 1675

Mathematical analysis is central to mathematics, whether pure or applied. This discipline arises in various mathematical models whose dependent variables vary continuously and are functions of one or several variables. Real analysis dates to the mid-nineteenth century, and its roots go back to the pioneering papers by Cauchy, Riemann, and Weierstrass.

In 1821, Cauchy established new requirements of rigor in his celebrated Cours d’Analyse . The questions he raised are the following:

– What is a derivative really? Answer: a limit. – What is an integral really? Answer: a limit. – What is an infinite series really? Answer: a limit.

This leads to – What is a limit? Answer: a number. And, finally, the last question: – What is a number?

Weierstrass and his collaborators (Heine, Cantor) answered this question around 1870–1872. Our treatment in this volume is strongly related to the pioneering contributions in differential calculus by Newton, Leibniz, Descartes, and Euler in the seventeenth and eighteenth centuries, with mathematical rigor in the nineteenth century pro- moted by Cauchy, Weierstrass, and Peano . This presentation furthers modern directions in the integral calculus developed by Riemann and Darboux.

Due to the huge impact of mathematical analysis, we have intended in this book to build a bridge between ordinary high-school or undergraduate exercises and more difficult and abstract concepts or problems related to this field. We present in this volume an unusual collection of creative problems in elementary mathematical anal- ysis. We intend to develop some basic principles and solution techniques and to offer

a systematic illustration of how to organize the natural transition from problem- solving activity toward exploring, investigating, and discovering new results and properties.

x Preface The aim of this volume in elementary mathematical analysis is to introduce,

through problems-solving, fundamental ideas and methods without losing sight of the context in which they first developed and the role they play in science and partic- ularly in physics and other applied sciences. This volume aims at rapidly developing differential and integral calculus for real-valued functions of one real variable, giving relevance to the discussion of some differential equations and maximum prin- ciples.

The book is mainly geared toward students studying the basic principles of math- ematical analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam and other high-level mathematical contests. We also address this work to motivated high-school and undergraduate students. This volume is meant primarily for students in mathematics, physics, engineering, and computer science, but, not without authorial ambition, we believe it can be used by anyone who wants to learn elementary mathematical analysis by solving prob- lems. The book is also a must-have for instructors wishing to enrich their teach- ing with some carefully chosen problems and for individuals who are interested in solving difficult problems in mathematical analysis on the real axis. The volume is intended as a challenge to involve students as active participants in the course. To make our work self-contained, all chapters include basic definitions and properties. The problems are clustered by topic into eight chapters, each of them containing both sections of proposed problems with complete solutions and separate sections including auxiliary problems, their solutions being left to our readers. Throughout the book, students are encouraged to express their own ideas, solutions, generaliza- tions, conjectures, and conclusions.

The volume contains a comprehensive collection of challenging problems, our goal being twofold: first, to encourage the readers to move away from routine exercises and memorized algorithms toward creative solutions and nonstandard problem-solving techniques; and second, to help our readers to develop a host of new mathematical tools and strategies that will be useful beyond the classroom and in a number of applied disciplines. We include representative problems proposed at various national or international competitions, problems selected from prestigious mathematical journals, but also some original problems published in leading publi- cations. That is why most of the problems contained in this book are neither standard nor easy. The readers will find both classical topics of mathematical analysis on the real axis and modern ones. Additionally, historical comments and developments are presented throughout the book in order to stimulate further inquiry.

Traditionally, a rigorous first course or problem book in elementary mathematical analysis progresses in the following order:

Sequences Functions =⇒ Continuity =⇒ Differentiability =⇒ Integration Limits

Preface xi However, the historical development of these subjects occurred in reverse order: Archimedes

Newton (1665) Cauchy (1821) ⇐= Weierstrass (1872) ⇐= = Kepler (1615) Leibniz (1675) ⇐ Fermat (1638)

This book brings to life the connections among different areas of mathematical analysis and explains how various subject areas flow from one another. The vol- ume illustrates the richness of elementary mathematical analysis as one of the most classical fields in mathematics. The topic is revisited from the higher viewpoint of university mathematics, presenting a deeper understanding of familiar subjects and an introduction to new and exciting research fields, such as Ginzburg–Landau equa- tions, the maximum principle, singular differential and integral inequalities, and nonlinear differential equations.

The volume is divided into four parts, ten chapters, and two appendices, as follows:

Part I. Sequences, Series, and Limits Chapter 1. Sequences Chapter 2. Series Chapter 3. Limits of Functions

Part II. Qualitative Properties of Continuous and Differentiable Functions

Chapter 4. Continuity Chapter 5. Differentiability

Part III. Applications to Convex Functions and Optimization Chapter 6. Convex Functions Chapter 7. Inequalities and Extremum Problems

Part IV. Antiderivatives, Riemann Integrability, and Applications Chapter 8. Antiderivatives Chapter 9. Riemann Integrability Chapter 10. Applications of the Integral Calculus

Appendix A. Basic Elements of Set Theory Appendix B. Topology of the Real Line

Each chapter is divided into sections. Exercises, formulas, and figures are num- bered consecutively in each section, and we also indicate both the chapter and the section numbers. We have included at the beginning of chapters and sections quo- tations from the literature. They are intended to give the flavor of mathematics as

a science with a long history. This book also contains a rich glossary and index, as well as a list of abbreviations and notation.

xii Preface Key features of this volume:

– contains a collection of challenging problems in elementary mathematical analysis; – includes incisive explanations of every important idea and develops illuminating applications of many theorems, along with detailed solutions, suitable cross- references, specific how-to hints, and suggestions;

– is self-contained and assumes only a basic knowledge but opens the path to com- petitive research in the field; – uses competition-like problems as a platform for training typical inventive skills; – develops basic valuable techniques for solving problems in mathematical ana-

lysis on the real axis; – 38 carefully drawn figures support the understanding of analytic concepts; – includes interesting and valuable historical account of ideas and methods in

analysis; – contains excellent bibliography, glossary, and index.

The book has elementary prerequisites, and it is designed to be used for lecture courses on methodology of mathematical research or discovery in mathematics. This work is a first step toward developing connections between analysis and other math- ematical disciplines, as well as physics and engineering.

The background the student needs to read this book is quite modest. Anyone with elementary knowledge in calculus is well-prepared for almost everything to

be found here. Taking into account the rich introductory blurbs provided with each chapter, no particular prerequisites are necessary, even if a dose of mathematical so- phistication is needed. The book develops many results that are rarely seen, and even experienced readers are likely to find material that is challenging and informative.

Our vision throughout this volume is closely inspired by the following words of George P´olya [90] (1945) on the role of problems and discovery in mathematics: Infallible rules of discovery leading to the solution of all possible mathematical problems would be more desirable than the philosopher’s stone, vainly sought by all alchemists. The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea. Those of us who have little luck and less brain sometimes sit for decades. The fact seems to be, as Poincar´e observed, it is the man, not the method, that solves the problem.

Despite our best intentions, errors are sure to have slipped by us. Please let us know of any you find.

August 2008 Teodora-Liliana R˘adulescu Vicent¸iu R˘adulescu Titu Andreescu

Acknowledgments

We acknowledge, with unreserved gratitude, the crucial role of Professors Cather- ine Bandle, Wladimir-Georges Boskoff, Louis Funar, Patrizia Pucci, Richard Stong, and Michel Willem, who encouraged us to write a problem book on this subject. Our colleague and friend Professor Dorin Andrica has been very interested in this project and suggested some appropriate problems for this volume. We warmly thank Professors Ioan S¸erdean and Marian Tetiva for their kind support and useful discus- sions.

This volume was completed while Vicent¸iu R˘adulescu was visiting the Univer- sity of Ljubljana during July and September 2008 with a research position funded by the Slovenian Research Agency. He would like to thank Professor Duˇsan Repovˇs for the invitation and many constructive discussions.

We thank Dr. Nicolae Constantinescu and Dr. Mirel Cos¸ulschi for the profes- sional drawing of figures contained in this book.

We are greatly indebted to the anonymous referees for their careful reading of the manuscript and for numerous comments and suggestions. These precious con- structive remarks were very useful to us in the elaboration of the final version of this volume.

We are grateful to Ann Kostant, Springer editorial director for mathematics, for her efficient and enthusiastic help, as well as for numerous suggestions related to previous versions of this book. Our special thanks go also to Laura Held and to the other members of the editorial technical staff of Springer New York for the excellent quality of their work.

We are particularly grateful to copyeditor David Kramer for his guidance, thor- oughness and attention to detail.

V. R˘adulescu acknowledges the support received from the Romanian Research Council CNCSIS under Grant 55/2008 “Sisteme diferent¸iale ˆın analiza neliniar˘a s¸i aplicat¸ii.”

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Abbreviations and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Part I Sequences, Series, and Limits

1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Main Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Introductory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Recurrent Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5 Hardy’s and Carleman’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.6 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.1 Main Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2 Elementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.3 Convergent and Divergent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.4 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.5 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.6 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.1 Main Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.2 Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.3 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.4 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.4 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Part II Qualitative Properties of Continuous and Differentiable Functions

4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.1 The Concept of Continuity and Basic Properties . . . . . . . . . . . . . . . . . 139

4.2 Elementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.3 The Intermediate Value Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.4 Types of Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.5 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.6 Functional Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.7 Qualitative Properties of Continuous Functions . . . . . . . . . . . . . . . . . . 169

4.8 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.1 The Concept of Derivative and Basic Properties . . . . . . . . . . . . . . . . . 183

5.2 Introductory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.3 The Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.4 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

5.5 Differential Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.6 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Part III Applications to Convex Functions and Optimization

6 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.1 Main Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.2 Basic Properties of Convex Functions and Applications . . . . . . . . . . . 265

6.3 Convexity versus Continuity and Differentiability . . . . . . . . . . . . . . . 273

6.4 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

6.5 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

7 Inequalities and Extremum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

7.1 Basic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

7.2 Elementary Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.3 Jensen, Young, H¨older, Minkowski, and Beyond . . . . . . . . . . . . . . . . . 294

7.4 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

7.5 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

7.6 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Part IV Antiderivatives, Riemann Integrability, and Applications

8 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

8.1 Main Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

8.2 Elementary Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

8.3 Existence or Nonexistence of Antiderivatives . . . . . . . . . . . . . . . . . . . 317

8.4 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.5 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Contents xvii

9 Riemann Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

9.1 Main Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

9.2 Elementary Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

9.3 Classes of Riemann Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . 337

9.4 Basic Rules for Computing Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 339

9.5 Riemann Iintegrals and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

9.6 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

9.7 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

10 Applications of the Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

10.2 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

10.3 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

10.4 Integrals and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

10.5 Applications to Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

10.6 Independent Study Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Part V Appendix

A Basic Elements of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

A.1 Direct and Inverse Image of a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

A.2 Finite, Countable, and Uncountable Sets . . . . . . . . . . . . . . . . . . . . . . . 418

B Topology of the Real Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

B.1 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

B.2 Some Distinguished Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Abbreviations and Notation

Abbreviations

We have tried to avoid using nonstandard abbreviations as much as possible. Other abbreviations include:

AMM American Mathematical Monthly GMA

Mathematics Gazette, Series A MM

Mathematics Magazine IMO

International Mathematical Olympiad IMCUS

International Mathematics Competition for University Students MSC

Mikl´os Schweitzer Competitions Putnam

The William Lowell Putnam Mathematical Competition

SEEMOUS South Eastern European Mathematical Olympiad for University Students

Notation

We assume familiarity with standard elementary notation of set theory, logic, algebra, analysis, number theory, and combinatorics. The following is notation that deserves additional clarification.

N the set of nonnegative integers (N = {0,1,2,3,...}) N ∗

the set of positive integers (N ∗ = {1,2,3,...}) Z

the set of integer real numbers (Z = {...,−3,−2,−1,0,1,2,3,...}) Z ∗

the set of nonzero integer real numbers (Z ∗ = Z \ {0}) Q

the set of rational real numbers

n ;m ∈ Z, n ∈ N ∗ , m and n are relatively prime

R the set of real numbers R ∗

the set of nonzero real numbers (R ∗ = R \ {0}) R +

the set of nonnegative real numbers (R + = [0, +∞)) R

the completed real line the completed real line

C the set of complex numbers

e n lim n → ∞

sup A the least upper bound of the set A ⊂R inf A

the greatest lower bound of the set A ⊂R x +

the positive part of the real number x (x + = max{x,0}) x −

the negative part of the real number x (x − = max{−x,0}) |x|

the modulus (absolute value) of the real number x ( |x| = x + +x − ) {x}

the fractional part of the real number x (x = [x] + {x}) Card (A)

cardinality of the finite set A dist (x, A) the distance from x ∈ R to the set A ⊂ R (dist(x,A) = inf{|x − a|; a ∈ A}) IntA

the set of interior points of A ⊂R

f (A) the image of the set A under a mapping f

f −1 (B) the inverse image of the set B under a mapping f

f ◦g the composition of functions f and g: ( f ◦ g)(x) = f (g(x)) n !

n factorial, equal to n (n − 1)···1

(n ∈ N ∗ )

(2n)!! 2n (2n − 2)(2n − 4)···4 · 2

(n ∈N ∗ )

(2n + 1)!! (2n + 1)(2n − 1)(2n − 3)···3 · 1

f (n) (x) n th derivative of the function f at x

C n (a, b) the set of n-times differentiable functions f : (a, b)→R such that f (n) is continuous on (a, b)

C ∞ (a, b) the set of infinitely differentiable functions f : (a, b)→R (C ∞

(a, b) = ∞

n =0 C (a, b))

Δ f the Laplace operator applied to the function f : D

⊂R →R

Landau’s notation f (x) = o(g(x)) as x→x 0 if f (x)/g(x)→0 as x→x 0

f (x) = O(g(x)) as x→x 0 if f (x)/g(x) is bounded in a neighborhood of x 0

f ∼ g as x→x 0 if f (x)/g(x)→1 as x→x 0

Hardy’s notation f ≺≺ g as x→x 0 if f (x)/g(x)→0 as x→x 0

f 0 if f (x)/g(x) is bounded in a neighborhood of x 0

Chapter 1 Sequences

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. — Albert Einstein (1879–1955)

Abstract. In this chapter we study real sequences, a special class of functions whose domain is the set N of natural numbers and range a set of real numbers.

1.1 Main Definitions and Basic Results

Hypotheses non fingo. [“I frame no hypotheses.”]

Sir Isaac Newton (1642–1727) Sequences describe wide classes of discrete processes arising in various applica-

tions. The theory of sequences is also viewed as a preliminary step in the attempt to model continuous phenomena in nature. Since ancient times, mathematicians have realized that it is difficult to reconcile the discrete with the continuous. We under- stand counting 1 , 2, 3, . . . up to arbitrarily large numbers, but do we also under- stand moving from 0 to 1 through the continuum of points between them? Around 450 BC, Zeno thought not, because continuous motion involves infinity in an essential way. As he put it in his paradox of dichotomy: There is no motion because that which is moved must arrive at the middle (of its course) before it arrives at the end.

Aristotle, Physics, Book VI, Ch. 9

A sequence of real numbers is a function f : N →R (or f : N ∗ →R). We usually write a n (or b n ,x n , etc.) instead of f (n). If (a n ) n ≥1 is a sequence of real numbers and if n 1 <n 2 < ··· < n k < ··· is an increasing sequence of positive integers, then the sequence (a n k ) k ≥1 is called a subsequence of (a n ) n ≥1 .

A sequence of real numbers (a ) nn ≥1 is said to be nondecreasing (resp., increas- ing ) if a n ≤a n +1 (resp., a n <a n +1 ), for all n ≥ 1. The sequence (a n ) n ≥1 is called nonincreasing (resp., decreasing) if the above inequalities hold with “ ≥” (resp., “ >”) instead of “≤” (resp., “<”).

4 1 Sequences y

1 y =l+ε

y =l−ε

x Fig. 1.1 Adapted from G.H. Hardy, Pure Mathematics, Cambridge University Press, 1952.

We recall in what follows some basic definitions and properties related to sequences. One of the main notions we need in the sequel is that of convergence. Let (a n ) n ≥1

be a sequence of real numbers. We say that (a n ) n ≥1 is a convergent sequence if there exists ℓ ∈ R (which is called the limit of (a n ) n ≥1 ) such that for each neighborhood N of ℓ, we have a n ∈ N for all n ≥ N, where N is a positive integer depending on N . In other words, (a n ) n ≥1 converges to ℓ if and only if for each ε > 0 there exists a natural number N = N( ε ) such that |a n − ℓ| < ε , for all

n ≥ N. In this case we write lim n → ∞ a n = ℓ or a n →ℓ as n→∞. The concentration of

a n ’s in the strip (ℓ − ε ,ℓ+ ε ) for n ≥ N is depicted in Figure 1.1. Example. Define a n = (1 + 1/n) n ,b n = (1 + 1/n) n +1 ,c n =∑ n k =0 1 /k!. Then (a n ) n ≥1 and (c n ) n ≥1 are increasing sequences, while (b n ) n ≥1 is a decreasing seq- uence. However, all these three sequences have the same limit, which is denoted by

e (e = 2.71828 . . .). We will prove in the next chapter that e is an irrational number. Another important example is given by the following formula due to Stirling, which

asserts that, asymptotically, n! behaves like n n e −n

2 π n . More precisely,

Stirling’s formula is important in calculating limits, because without this asymptotic property it is difficult to estimate the size of n! for large n. In this capacity, it plays an important role in probability theory, when it is used in computing the probable outcome of an event after a very large number of trials. We refer to Chapter 9 for a complete proof of Stirling’s formula.

We say that the sequence of real numbers (a n ) n ≥1 has limit +∞ (resp., −∞) if to each α ∈ R there corresponds some positive integer N 0 such that n ≥N 0 implies

a n > α (resp., a n < α ). We do not say that (a n ) n ≥1 converges in these cases. An element α ∈ R is called an accumulation point of the sequence (a n ) n ≥1 ⊂R if there exists a subsequence (a n k ) k ≥1 such that a n k → α as n k →∞. Any convergent sequence is bounded. The converse is not true (find a counter- example!), but we still have a partial positive answer, which is contained in the following result, which is due to Bernard Bolzano (1781–1848) and

1.1 Main Definitions and Basic Results 5 Karl Weierstrass (1815–1897). It seems that this theorem was revealed for the first

time in Weierstrass’s lecture of 1874.

A bounded sequence of real numbers has a convergent subsequence. However, some additional assumptions can guarantee that a bounded sequence converges. An important criterion for convergence is stated in what follows. Monotone Convergence Theorem. Let (a n ) n ≥1

Bolzano–Weierstrass Theorem.

be a bounded sequence that is monotone. Then (a n ) n ≥1 is a convergent sequence. If increasing, then lim n → ∞ a n =

sup n a n , and if decreasing, then lim n → ∞ a n = inf n a n .

In many arguments a central role is played by the following principle due to Cantor. Nested Intervals Theorem. Suppose that I n = [a n ,b n ] are closed intervals such that I n +1 ⊂I n , for all n ≥1 . If lim n → ∞ (b n −a n )=0 , then there is a unique real

number x 0 that belongs to every I n .

A sequence (a n ) n ≥1 of real numbers is called a Cauchy sequence [Augustin Louis Cauchy (1789–1857)] if for every ε > 0 there is a natural number N ε such that |a m −a n |< ε , for all m ,n≥N ε . A useful result is that any Cauchy sequence is a bounded sequence. The following strong convergence criterion reduces the study of convergent sequences to that of Cauchy sequences.

A sequence of real numbers is convergent if and only if it is a Cauchy sequence. Let (a n ) n ≥1

Cauchy’s Criterion.

be an arbitrary sequence of real numbers. The limit inferior of (a n ) n ≥1 (denoted by lim inf n → ∞ a n ) is the supremum of the set X of x ∈ R such

that there are at most a finite numbers of n ∈N ∗ for which a n < x. The limit su- perior of (a n ) n ≥1 (denoted by lim sup n → ∞ a n ) is the infimum of the set Y of y ∈R such that there are finitely many positive integers n for which a n > y. Equivalent characterizations of these notions are the following:

(i) ℓ = lim sup n → ∞ a n ( ℓ ∈ R) if and only if whenever α < ℓ the set {n ≥ N ∗ ;a n > α } is infinite, and whenever ℓ < β the set {n ≥ N ∗ ;a n > β } is finite; (ii) ℓ = lim sup n → ∞ a n if and only if ℓ = inf m ≥1 sup n ≥m a n .

Exercise. Formulate the above characterizations for lim inf n → ∞ a n . The limit inferior and the limit superior of a sequence always exist, possibly in R. Moreover, the following relations hold:

∞ n ≤ limsup a n → , n → ∞ lim inf n → ∞ (a n +b n ) ≥ liminf n

lim inf a

→ ∞ a n + lim inf n → ∞ b n , lim sup (a n +b n ) ≤ limsup a n + lim sup b n .

The existence of the limit of a sequence is closely related to “lim inf” and “lim sup.” More precisely, the sequence (a n ) n ≥1 has a limit if and only if

lim inf n → ∞ a n = lim sup n → ∞ a n .

A very useful result in applications concerns the following link between the ratio and the nth root of the terms of a sequence of positive numbers.

6 1 Sequences

Theorem. Let (a n ) n ≥1

be a sequence of positive numbers. Then

a n +1

lim inf +1 n a n

≤ liminf ∞ a n ≤ limsup a n ≤ limsup . n →

The above result implies, in particular, that if lim n ∞ (a n +1 /a n ) = ℓ, with 0 ≤ ℓ ≤ ∞, then lim n ∞

a n = ℓ. Example. The above theorem implies that lim

√ ∞ a = 1, where a = n n → n n n . We

can give an elementary proof to the fact that a n >a n +1 , for all n ≥ 3. Indeed, define

b n = (a

n /a n +1 ) n , for all n ≥ 2. Then b n >b n −1 for n ≥ 2 is equivalent to n 2 =

(n+1)

a 2n >a n +1 n −1

n +1 a n −1 =n − 1. It remains to show that b n > 1 for all n ≥ 3. This follows from the fact that (b n ) n is increasing, combined with b 3 =3 ≥2 4 /4 3 = 81/64 > 1.

A sequence that is not convergent is called a divergent sequence. An important example of a divergent sequence is given by a n = 1 + 1/2 + 1/3 + ···+ 1/n, n ≥ 1. Indeed, since (a n ) n ≥1 is increasing, it has a limit ℓ ∈ R ∪ {+∞}. Assuming that ℓ is finite, it follows that

2 4 6 + ··· = 2 2 3 + ··· → 2 →∞. This means that

which is impossible because 1 /1 > 1/2, 1/3 > 1/4, 1/5 > 1/6, and so on. Con- sequently, a n → + ∞ as n→∞. The associated series 1 + 1/2 + 1/3 + 1/4 + ··· is usually called the harmonic series. Furthermore, if p < 1 we have 1/n p > 1/n, for

all n ≥ 2. It follows that the sequence (a n ) n ≥1 defined by a n =∑ n k =1 1 /k p diverges, too. The situation is different if p > 1. In this case, the sequence (a n ) n ≥1 defined as above is convergent. Indeed,

(2n) p

4 p (2n) p

=1 + p +

<1+2 1 −p a 2n +1 ,

because a n <a 2n +1 . Thus (1 − 2 1 −p )a 2n +1 < 1. Since p > 1, we have 1 − 2 1 −p > 0. Hence a 2n +1 < (1 − 2 1 −p ) −1 for all n ≥ 1. So, the sequence (a n ) n ≥1 (for p > 1) is increasing and bounded above by (1 − 2 1 −p ) −1 , which means that it converges. The corresponding limit lim

n → ∞ ∑ n k =1 1 /k p =∑ n =1 1 /n p is denoted by ζ (p) (p > 1) and is called the Riemann zeta function [Georg Friedrich Bernhard Riemann (1826–1866)]. This function is related to a celebrated mathematical conjecture (Hilbert’s eighth problem). The Riemann hypothesis, formulated in 1859, is closely related to the frequency of prime numbers and is one of the seven grand challenges

1.2 Introductory Problems 7 of mathematics (“Millennium Problems,” as designated by the Clay Mathematics

Institute; see http://www.claymath.org/millennium/). The following result (whose “continuous” variant is l’Hˆopital’s rule for differen- tiable functions) provides us a method to compute limits of the indeterminate form

0 /0 or ∞/∞. This property is due to Otto Stolz (1859–1906) and Ernesto Ces`aro (1859–1906). Stolz–Ces`aro Lemma. Let (a n ) n ≥1 and (b n ) n ≥1

be two sequences of real num- bers.

(i) Assume that a n →0 and b n →0 as n →∞ . Suppose, moreover, that (b n ) n ≥1 is decreasing for all sufficiently large n and there exists

Then there exists lim n → ∞ a n /b n and, moreover, lim n → ∞ a n /b n =ℓ . (ii) Assume that b n →+ ∞ as n →∞ and that (b n ) n ≥1 is increasing for all sufficiently

large n . Suppose that there exists

a lim n +1 −a n

n → ∞ b n =: ℓ ∈ R. +1 −b n

Then there exists lim n → ∞ a n /b n and, moreover, lim n → ∞ a n /b n =ℓ .

1.2 Introductory Problems

Nature not only suggests to us problems, she suggests their solution.

Henri Poincar´e (1854–1912) We first prove with elementary arguments the following basic result.

Arithmetic–Geometric Means (AM–GM) Inequality. For any positive num- bers a 1 , a 2 ,...,a n we have

a 1 +a 2 + ··· + a n

≥ a 1 a 2 ··· a n n . Replacing a k with a √ k n a 1 ···a n (for 1 ≤ k ≤ n), we have a 1 a 2 ···a n = 1, so it

is enough to prove that a 1 +a 2 + ··· + a n ≥ n. We argue by induction. For n = 1 the property is obvious. Passing from n to n + 1, we can assume that a 1 ≤1≤a 2 . Hence (1 − a 1 )(a 2 − 1) ≥ 1, that is, a 1 +a 2 ≥1+a 1 a 2 . Next, we set b 1 =a 1 a 2 and, for any k = 2, . . . , n, we set b k =a k +1 . Hence b 1 b 2 ··· b n = 1. So, by the induction hypothesis,

a 1 a 2 +a 3 + ··· + a n ≥ n.

8 1 Sequences It follows that

a 1 +a 2 + ··· + a n +1 ≥a 1 a 2 +1+a 3 + ··· + a n +1 ≥ n + 1. The AM–GM inequality also implies a relationship between the harmonic mean

and the geometric mean: for any positive numbers a 1 ,a 2 ,...,a n we have

1 1 1 ≤ a 1 a 2 ··· a n .

a 1 + a 2 + ··· + a n

The following easy exercise shows the importance of elementary monotony argu- ments for deducing the value of the limit of a convergent sequence.

1.2.1. Let A 1 ,A 2 ,...,A k

be nonnegative numbers. Compute

Solution. Without loss of generality, we may assume that A 1 = max{A 1 ,...,A k }.

Therefore A n 1 ≤A n 1 + ··· + A n k ≤ kA n 1 . It follows that

which shows that the limit sought is A 1 : = min{A j ;1 ≤ j ≤ k}. ⊓ ⊔ Comments. We easily observe that the above result does not remain true if we

do not assume that the numbers A 1 ,...,A k are nonnegative. Moreover, in such a case it is possible that the sequence defined by B n : = n 1 + ··· + A n 1 k /n is not even

convergent (give an example!). We have already provided some basic examples of convergent and divergent

sequences. The next exercise shows how, by means of monotony principles, we can construct further examples of such sequences.

1.2.2. Prove that lim n → ∞ n ! =+∞ and lim n → ∞ n ! /n = e −1 . Solution. We first observe that (2n)! ≥ ∏ 2n

(2n + 1)! ≥ n n +1 ≥ n . For the last part we take into account that for any sequence (a n ) n ≥1 of positive

numbers we have

a n +1 lim inf

a n +1

≤ liminf a n ≤ limsup

a n ≤ limsup

√ = e, we conclude that lim n n → ∞ n ! /n = e −1 . An alternative argument is based on the Stolz–Ces`aro lemma

1 Taking a /n n = n!/n n and using lim n → ∞

applied to a n = log(n!/n n ) and b n = n and using again lim n → ∞ −1 1 /n = e. ⊓ ⊔

1.2 Introductory Problems 9

A nonobvious generalization of the above property is stated below. Our proof applies subtle properties of real-valued functions (see Chapter 5) but we strongly suggest that the reader refine the monotony arguments developed above.

Independent Study. Prove that for real number p ≥0 we have

n p 1 /n p ·2 +1 ···n

. Particular case:

lim =e −1/(p+1) n 2

n 1 /(p+1)

Hint. Use the mean value theorem. For instance, in the particular case p = 2, apply the Lagrange mean value theorem to the function f (x) = (x 3 ln x

)/3 −x 3 /9 on the interval [k, k + 1], 1 ≤ k ≤ n.

We have seen above that n ! →∞ as n→∞. A natural question is to study the asymptotic behavior of the difference of two consecutive terms of this sequence. The next problem was published in 1901 and is due to the Romanian mathemati- cian Traian Lalescu (1882–1929), who wrote one of the first treatises on integral equations.

1.2.3. Find the limit of the sequence (a n n ) n +1 ≥2 defined by a n = (n + 1)!− n ! .

√ n Solution. We can write a n = n !

(b n n − 1), where b n = +1 (n + 1)!/ n !. Hence

n n → ∞ n ! /n = e −1 , so b n →1 as n→∞. On the other hand,

So, by (1.1) and (1.2), we obtain that a n →e −1 as n →∞. ⊓ ⊔ Remark. In the above solution we have used the property that if b n →1 as n→∞

then (b n − 1)/lnb n →1 as n→∞. This follows directly either by applying the Stolz– Ces`aro lemma or after observing that

ln e Independent Study. Let p

be a nonnegative real number. Study the convergence of the sequence (x n ) n ≥1 defined by

n p 1 /n n

pp

1 /(n+1) p 1 +1 2 (n+1)

p +1

·2 ··· (n + 1)

·2 ··· n .

10 1 Sequences We already know that the sequence (s n ) n ≥1 defined by s n = 1 + 1/2 + ···+ 1/n

diverges to + ∞. In what follows we establish the asymptotic behavior for s n /n as n

→∞. As a consequence, the result below implies that lim n → ∞ n (1 − n ) = ∞.

1.2.4. Consider the sequence (s n ) n ≥1 defined by s n = 1 + 1/2 + ···+ 1/n . Prove that

(a) n (n + 1) 1 /n <n+s n , for all integers n >1 ; (b) (n − 1)n −1/(n−1) <n−s n , for all integers n >2 .

Solution. (a) Using the AM–GM inequality we obtain

(b) By the definition of (s n ) n ≥1 we have n

1 1 −s 1

1 /(n−1) =

··· 1 − =n −1/(n−1) . n −1

n −1

The following exercise involves a second-order linear recurrence.

1.2.5. Let α ∈ (0,2) . Consider the sequence defined by

x n +1 = α x n + (1 − α )x n −1 , for all n ≥ 1. Find the limit of the sequence in terms of α, x 0 , and x 1 .

Solution. We have x n −x n −1 =( α − 1)(x n −x n −1 ). It follows that x n −x n −1 = ( α − 1) n −1 (x 1 −x 0 ). Therefore

x n −x 0 = ∑ (x

k −x k −1

) = (x

1 −x 0 ∑ −1 − 1) .

k =1

k =1

Now, using the assumption α ∈ (0,2), we deduce that

(1 − α lim )x x 0 +x 1

Next, we discuss a first-order quadratic recurrence in order to establish a neces- sary and sufficient condition for convergence in terms of the involved real parameter.

1.2.6. Let a be a positive number. Define the sequence (x n ) n ≥0 by x n +1 =a+x 2 n , for all n ≥ 0, x 0 = 0.

1.2 Introductory Problems 11

Find a necessary and sufficient condition such that the sequence is convergent. Solution. If lim n ∞ x n = ℓ then ℓ = a + ℓ 2 → , that is,

1 ± 1 − 4a

So, necessarily, a ≤ 1/4. Conversely, assume that 0 < a ≤ 1/4. From x n +1 −x n =x 2 −x n 2 n −1 it follows that the sequence (x n ) is increasing. Moreover,

n +1 =a+x n < + = , 4 4 2

provided that x n < 1/2. This shows that the sequence is bounded, so it converges. ⊓ ⊔

The above result is extended below to larger classes of quadratic nonlinearities. The close relationship between boundedness, monotony, and convergence is pointed out and a complete discussion is developed in the following exercise.

1.2.7. Let f (x) = 1/4 + x − x 2 . For any x ∈R , define the sequence (x n ) n ≥0 by x 0 =x and x n +1 = f (x n ) . If this sequence is convergent, let x ∞

be its limit. (a) Show that if x =0 , then the sequence is bounded and increasing, and compute

its limit x ∞ =ℓ . (b) Find all possible values of ℓ and the corresponding real numbers x such that x ∞ =ℓ .

Solution. (a) We have

f (x) = − x −

so x n ≤ 1/2, for all n ≥ 1. This inequality also shows that (x n ) n ≥0 is increasing. Passing to the limit, we obtain

Since all the terms of the sequence are positive, we deduce that ℓ = 1/2. (b) By the definition of f it follows that

So, the sequence diverges both if x ≤ −1/2 and for x ≥ 3/2.

12 1 Sequences We now prove that if x ∈ (−1/2,3/2), then the sequence converges and its limit

equals 1 /2. Indeed, in this case we have

f (x) −

It follows that

x n +1 −

→0 as n→∞. ⊓ ⊔

The next exercise gives an example of a convergent sequence defined by means of an integer-valued function. We invite the reader to establish more properties of the function

be the closest integer to √ n . Compute

1.2.8. For any integer n ≥1 , let

n 2 +2

lim

n → ∞ ∑ j =1 2 j .

Solution. Since (k − 1/2) 2 =k 2 − k + 1/4 and (k + 1/2) 2 =k 2 + k + 1/4, it fol- lows that

2 −k+1≤n≤k 2 + k. Hence n 2 +2

k 2 +k

2 k +2 −k

n lim → ∞ ∑ 2 j = ∑ ∑

n j =1

k =1 k =1 n =k 2 −k+1

= ∑ (2 k +2 −k )(2 −k +k −2 −k −k )= (2 −k(k−2) −2 −k(k+2) ∑ )

k =1 k =1 ∞

= ∑ 2 −k(k−2) − ∑ 2 −k(k−2) = 3. ⊓ ⊔

k =1

k =3

We give below a characterization of the sequences having a certain growth prop- erty. As in many cases, monotony arguments play a central role.

be a sequence of real numbers such that a 1 =1 and a n +1 > 3a n /2 for all n ≥1 . Prove that the sequence (b n ) n ≥1 defined by

1.2.9. (i) Let (a n ) n ≥1

b n = (3/2) n −1

either has a finite limit or tends to infinity. (ii) Prove that for all α >1 there exists a sequence (a n ) n ≥1 with the same properties

such that

n → ∞ (3/2) n −1

lim

International Mathematical Competition for University Students, 2003 Solution. (i) Our hypothesis a n +1 > 3a n /2 is equivalent to b n +1 >b n , and the

conclusion follows immediately.

1.2 Introductory Problems 13

(ii) For any α > 1 there exists a sequence 1 = b 1 <b 2 < ··· that converges to α . Choosing a n = (3/2) n −1 b n , we obtain the required sequence (a n ) n ≥1 . ⊓ ⊔

Qualitative properties of a sequence of positive integers are established in the next example.

1.2.10. Consider the sequence (a n ) n ≥1 defined by a n =n 2 +2 . (i) Find a subsequence n k k ≥1 such that if i

(ii) Find a subsequence such that any two terms are relatively prime. Solution. (i) For any integer k n (3 k +1)/2 3 ≥ 1, take n

. Then a n k =2 +1 and the conclusion follows.

k =2

(ii) Consider the subsequence (b n ) n ≥1 defined by b 1 = 3, b 2 =b 2 1 + 2, and, for any

integer n ≥ 3, b n = (b 1 ··· b n −1 ) 2 + 2. ⊓ ⊔

The Stolz–Ces`aro lemma is a powerful instrument for computing limits of seq- uences (always keep in mind that it gives only a sufficient condition for the exis- tence of the limit!). A simple illustration is given in what follows.

1.2.11. The sequence of real numbers (x n ) n ≥1 satisfies lim n → ∞ (x 2n +x 2n +1 )= 315 and lim n → ∞ (x 2n +x 2n −1 ) = 2003 . Evaluate lim n → ∞ (x 2n /x 2n +1 ) .

Harvard–MIT Mathematics Tournament, 2003 Solution. Set a n =x 2n and b n =x 2n +1 and observe that

b n +1 −b n (x 2n +3 +x 2n +2 ) − (x 2n +2 +x 2n +1 ) 315 − 2003 as n →∞. Thus, by the Stolz–Ces`aro lemma, the required limit equals −1. ⊓ ⊔

Remark. We observe that the value of lim n → ∞ (x 2n /x 2n +1 ) does not depend on the values of lim n → ∞ (x 2n +x 2n +1 ) and lim n → ∞ (x 2n +x 2n −1 ) but only on the con- vergence of these two sequences.

We refine below the asymptotic behavior of a sequence converging to zero. The proof relies again on the Stolz–Ces`aro lemma, and the method can be extended to large classes of recurrent sequences.

1.2.12. Let (a n ) n ≥1 n be a sequence of real numbers such that lim n → ∞ a n ∑

k =1

a 2 k =1 . Prove that lim n → ∞ (3n) 1 /3 a n =1 .

I. J. Schoenberg, Amer. Math. Monthly, Problem 6376 Solution. Set s n =∑ n k =1 a 2 k . Then the condition a n s n →1 implies that s n →∞ and

a n →0 as n→∞. Hence we also have that a n s n −1 →1 as n→∞. Therefore

s 3 3 2 2 n 2 −s n −1 =a n (s n +s n s n −1 +s n −1 )→3 as n→∞.

(1.3) Thus, by the Stolz–Ces`aro lemma, s 3 n /n→3 as n→∞, or equivalently, lim n → ∞ n −2/3

s 2 n =3 2 /3 . So, by (1.3), we deduce that lim n → ∞ (3n) 1 /3 a n = 1. ⊓ ⊔

14 1 Sequences We study in what follows a sequence whose terms are related to the coefficients

of certain polynomials.

1.2.13. Fix a real number x

2 . Consider the sequence (s n ) n ≥1 defined by s n =∑ n k =0 a k x k such that lim n → ∞ s

n = 1/(1 − 2x − x ) . Prove that for any integer

n ≥0 , there exists an integer m such that a 2 n +a 2 n +1 =a m .

Solution. We have

∑ k (1 ± 2 ) x k .

A straightforward computation shows that a 2 n 2 +a n +1 =a 2n +2 . ⊓ ⊔ The following (not easy!) problem circulated in the folklore of contestants in

Romanian mathematical competitions in the 1980s. It gives us an interesting prop- erty related to bounded sequences of real numbers. Does the property given below remain true if b n are not real numbers?

1.2.14. Suppose that (a n ) n ≥1 is a sequence of real numbers such that lim n → ∞ a n =1 and (b n ) n ≥1 is a bounded sequence of real numbers. If k is a positive integer such that lim n → ∞ (b n −a n b n +k )=ℓ , prove that ℓ=0 .

Solution. (C˘alin Popescu). Let b = lim inf n → ∞ b n and B = lim sup n → ∞ b n . Since (b n ) n ≥1 is bounded, both b and B are finite. Now there are two subsequences (b p r ) and (b q r ) of (b n ) n ≥1 such that b p r → b and b q r → B as r → ∞. Since a n → 1 and

b n −a n b n +k → ℓ as n → ∞, it follows that the subsequences (b p r +k ) and (b q r +k ) of (b n ) n ≥1 tend to b − ℓ and B − ℓ, respectively, as r → ∞. Consequently, b − ℓ ≥ b and

B − ℓ ≤ B, hence ℓ = 0. ⊓ ⊔ Elementary trigonometry formulas enable us to show in what follows that a very

simple sequence diverges. A deeper property of this sequence will be proved in Problem 1.4.26.