Selected Titles in This Series

Volume

4 W. J. Kaczor and M. T. Nowak Problems in mathematical analysis I: Real numbers, sequences and series

2000

3 Roger Knobel

An introduction to the mathematical theory of waves

2000

2 Gregory F. Lawler and Lester N. Coyle

Lectures on contemporary probability

1999

1 Charles Radin

Miles of tiles

1999

http://dx.doi.org/10.1090/stml/004

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Problems in Mathematical Analysis I Real Numbers, Sequences and Series

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STUDENT MATHEMATICAL LIBRARY Volume 4

Problems in Mathematical Analysis I Real Numbers, Sequences and Series

W J. Kaczor M.T. Nowak

iAMS AMERICAN MATHEMATICAL SOCIETY

Editorial Board David Bressoud Car l Pomerance Rober t Devaney, Chair Hung-Hsi Wu

Originally published in Polish, as Zadania z Analizy Matematyczne j . Cz$sc Pierwsza.

Liczby Rzeczywiste, Ciagi i Szeregi Liczbowe

© 1996, Wydawnictwo Uniwersyte tu Mari i Curie-Sklodowskiej , Lublin.

Translated, revised and augmented by t h e au thors .

2000 Mathematics Subject Classification. Primary 00A07, 40-01.

Library of Congress Cataloging-in-Publicat ion D a t a

Kaczor, W. J. (Wieslawa J.), 1949-[Zadania z analizy matematycznej. English] Problems in mathematical analysis. I. Real numbers, sequences and series /

W. J. Kaczor, M. T. Nowak. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 4)

Includes bibliographical references. ISBN 0-8218-2050-8 (softcover : alk. paper) 1. Mathematical analysis. I. Nowak, M. T. (Maria T.), 1951- II. Title.

III. Series. QA300K32513 2000 515/.076-dc21 99-087039

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org.

© 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at URL: http://www.ams.org/

10 9 8 7 6 5 4 3 2 09

Contents

Preface

Notation and Terminology

Problems

Chapter 1. Real Numbers

1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions

1.2. Some Elementary Inequalities

Chapter 2. Sequences of Real Numbers

2.1. Monotonic Sequences

2.2. Limits. Properties of Convergent Sequences

2.3. The Toeplitz Transformation, the Stolz Theorem and their Applications

2.4. Limit Points. Limit Superior and Limit Inferior

2.5. Miscellaneous Problems

Chapter 3. Series of Real Numbers

3.1. Summation of Series

Vlll Contents

3.2. Series of Nonnegative Terms 72

3.3. The Integral Test 88

3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz 92

3.5. The Dirichlet and Abel Tests 99

3.6. Cauchy Product of Infinite Series 102

3.7. Rearrangement of Series. Double Series 105

3.8. Infinite Products 112

Solutions

Chapter 1. Real Numbers

1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions 125

1.2. Some Elementary Inequalities 136

Chapter 2. Sequences of Real Numbers

2.1. Monotonic Sequences 151

2.2. Limits. Properties of Convergent Sequences 162

2.3. The Toeplitz Transformation, the Stolz Theorem and their Applications 181

2.4. Limit Points. Limit Superior and Limit Inferior 189

2.5. Miscellaneous Problems 208

Chapter 3. Series of Real Numbers

3.1. Summation of Series 245

3.2. Series of Nonnegative Terms 269

3.3. The Integral Test 302

3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz 309

3.5. The Dirichlet and Abel Tests 324

3.6. Cauchy Product of Infinite Series 333

Contents IX

3.7. Rearrangement of Series. Double Series 342

3.8. Infinite Products 360

Bibliography - Books 379

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Preface

This book is an enlarged and revised English edition of a Polish version published in 1996 by the Publishing House of Maria Curie-Sklodowska University in Lublin, Poland. It is the first volume of a planned series of books of problems in mathematical analysis. The second volume, already published in Polish, is under translation into English. The series is mainly intended for students who take courses in basic principles of analysis. The choice and arrangement of the material make it suitable for self-study, and instructors may find it useful as an aid in organizing tutorials and seminars.

This volume covers three topics: real numbers, sequences, and series. It does not contain problems concerning metric and topological spaces, which we intend to present in subsequent volumes.

The book is divided into two parts. The first part is a collection of exercises and problems, and the second contains their solutions. Complete solutions are given in most cases. Where no difficulties could be expected or when an analogous problem has already been solved, only a hint or simply an answer is given. Very often various solutions of a given problem are possible; we present here only one, hoping students themselves will find others.

XI

xi i Preface

With the student in mind, we have tried to keep things at an ele­mentary level whenever possible. For example, we present an elemen­tary proof of the Toeplitz theorem about the so-called regular trans­formation of sequences, which in many texts is proved by methods of functional analysis. The proof presented is taken from Toeplitz's orig­inal paper, published in 1911 in Prace Matematyczno-Fizyczne, Vol. 22. We hope that our presentation of this part of real analysis will be more accessible to readers and will ensure wider understanding.

All the notations and definitions used in this volume are standard and commonly used. The reader can find them, for example, in the textbooks [12] and [23], in which all necessary theoretical background can be found. However, to make the book consistent and to avoid ambiguity, a list of notations and definitions is included.

We have borrowed freely from many textbooks, problem books and problem sections of journals like the American Mathematical Monthly, Mathematics Today (Russian) and Delta (Polish). A com­plete list is given in the bibliography. It was beyond the authors' scope to trace all original sources, and we may have overlooked some contributions. If this has happened, we offer our sincere apologies.

We are deeply indebted to all our friends and colleagues from the Department of Mathematics of Maria Curie-Sklodowska University who offered stimulating suggestions. We have had many fruitful con­versations with M. Koter-Morgowska, T.Kuczumow, W. Rzymowski, S. Stachura and W. Zygmunt. Our sincere thanks are also due to Professor Jan Krzyz for his help in preparing the first version of the English manuscript. We are pleased to express our gratitude to Pro­fessor Kazimierz Goebel for his encouragement and active interest in the project. It is our pleasure to thank Professor Richard J. Libera, University of Delaware, for his invaluable and most generous help with the English translation and for all his suggestions and correc­tions which greatly improved the final version of the book.

W. J. Kaczor, M. T. Nowak

Notation and Terminology

• R - the set of all real numbers • R + - the set of all positive real numbers • Z - the set of all integers • N - the set of all positive integers • Q - the set of all rationals • (a, b) - open interval with the endpoints a and b • [a, b] - closed interval with the endpoints a and b • [x] - the integral part of a real number x • For x e R,

sgnx = <

( 1 for x > 0,

- 1 for x < 0,

0 for x = 0.

For n e N, n! = 1 - 2 - 3 - . . . - n , (2n)!! = 2 • 4 • 6 •... • (2n - 2)(2n) and (2n - 1)!! = 1 • 3 • 5 • ... • (2n - 3)(2n - 1). If A c K is nonempty and bounded from above, then sup A denotes the least upper bound of A. If a nonempty set A is not bounded above, then we assume that sup A = -foo.

xin

XIV Notation and Terminology

If A c t is nonempty and bounded from below, then inf A denotes the greatest lower bound of A. If a nonempty set A is not bounded below, then we assume that inf A = — oo. A sequence {an} of real numbers is said to be monotonically increasing (monotonically decreasing) if a n + i > an for all n £ N (an+i < an for all n £ N). The class of mono-tonic sequences consists of the increasing and the decreasing sequences. A number c is a limit point of the sequence {an} if there is a subsequence {ank} of {an} converging to c. Let S be the set of all the limit points of {an}. The limit infe­rior, lim an, and the limit superior, lim an, of the sequence

{an} are defined as follows:

+oc if {an} is not bounded above,

—oo if {an} is bounded above and S = 0,

supS if {an} is bounded above and S ^ 0,

-oo if {an} is not bounded below,

-f oo if {an} is bounded below and S = 0,

inf S if {an} is bounded below and S ^ 0.

lim an = <

lim an n—+oc

An infinite product Yl an is said to be convergent if there 7 1 = 1

exists no £ N such that an ^ 0 for n > UQ and the sequence {an oan o + i • . . . • anQ+n} converges, as n —> oo, to a limit P0

other than zero. The number P = a ^ • • • • * a>n0-i ' Po is called the value of the infinite product.

Bibliography - Books

References

[I] J. Banas, S. W§drychowicz, Zbior zadari z analizy matematycznej, Wy-dawnictwa Naukowo-Techniczne, Warszawa, 1994.

[2] W. I. Bernuk, I. K. Zuk, O.W. Melnikov, Sbornik olimpiadnych zadac po matematike, Narodnaja Asveta, Minsk, 1980.

[3] P. Biler, A. Witkowski, Problems in Mathematical Analysis, Marcel Dekker, Inc, New York and Basel, 1990.

[4] T. J. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan and Co., Limited, London, 1949.

[5] R. B. Burckel, An Introduction to Classical Complex Analysis, Aca­demic Press, New York San Francisco, 1979.

[6] B. P. Demidovic, Sbornik zadac i upraznenij po matematiceskomu anal-izu, Nauka, Moskva, 1969.

[7] A. J. Dorogovcev, Matematiceskij analiz. Spravocnoe posobe, Vyscaja Skola, Kiev, 1985.

[8] A. J. Dorogovcev, Matematiceskij analiz. Sbornik zadac, Vyscaja Skola, Kiev, 1987.

[9] G. M. Fichtenholz, Differential- und Integralrechnung, I, II, III, V.E.B. Deutscher Verlag Wiss., Berlin, 1966-1968.

[10] G. H. Hardy, A Course of Pure Mathematics, Cambridge University Press, Cambridge, 1946.

[II] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge Uni­versity Press, Cambridge, 1967.

379

380 Bibliography - Books

[12] G. Klambauer, Mathematical Analysis, Marcel Dekker, Inc., New York,

1975.

[13] G. Klambauer, Problems and Propositions in Analysis, Marcel Dekker, Inc., New York and Basel, 1979.

[14] K. Knopp, Theorie und Anwendung der Unendlichen Reihen, Springer-Verlag, Berlin and Heidelberg, 1947.

[15] L. D. Kudriavtsev, A. D. Kutasov, V. I. Chejlov, M. I. Shabunin, Problemas de Andlisis Matemdtico. Limite, Continuidad, Derivabili-dad, Mir, Moskva, 1989.

[16] K. Kuratowski, Introduction to Calculus, Pergamon Press, Oxford-Edinburgh-New York; Polish Scientific Publishers, Warsaw, 1969.

[17] D. S. Mitrinovic, Elementary Inequalities, P. Noordhoff Ltd., Gronin-gen, 1964.

[18] D. S. Mitrinovic, D. D. Adamovic, Nizovi i Redovi. Definicije, stavovi, zadaci, problemi (Serbo-Croatian), Naucna Knjiga, Belgrade, 1971.

[19] A. Ostrowski, Aufgabensammlung zur Infinitesimalrechnung, Band I: Funktionen einer Variablen, Birkhauser Verlag, Basel und Stuttgart, 1964.

[20] G. Polya, G. Szego, Problems and theorems in analysis I, Spriger-Verlag, Berlin Heidelberg New York, 1978.

[21] Ya. I. Rivkind, Zadaci po matematiceskomu analizu, Vysejsaja Skola, Minsk, 1973.

[22] W. I. Rozhkov, G. D. Kurdevanidze, N. G. Panfilov, Sbomik zadac matematiceskich olimpiad, Izdat. Univ. Druzhby Narodov, Moskva, 1987.

[23] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Company, New York, 1964.

[24] W. A. Sadownicij, A. S. Podkolzin, Zadaci studenceskich olimpiad po matematike, Nauka, Moskva, 1978.

[25] W. Sierpinski, Arytmetyka teoretyczna, PWN, Warszawa, 1959.

[26] W. Sierpinski, Dzialania nieskonczone, Czytelnik, Warszawa, 1948.

[27] H. Silverman, Complex variables, Houghton Mifflin Company, Boston, 1975.

[28] G. A. Tonojan, W. N. Sergeev, Studenceskije matematiceskije olimpia-dy, Izdatelstwo Erevanskogo Universiteta, Erivan, 1985.