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Problems in Mathematical Analysis I: Real Numbers, Sequences and Series PDF

396 Pages·2000·21.366 MB·English
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Preview Problems in Mathematical Analysis I: Real Numbers, Sequences and Series

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CCCooopppyyyrrriiiggghhhttt 222000000000 AAAmmmeeerrriiicccaaannn MMMaaattthhheeemmmaaatttiiicccaaalll SSSoooccciiieeetttyyy... DDDuuupppllliiicccaaatttiiiooonnn ppprrrooohhhiiibbbiiittteeeddd... PPPllleeeaaassseee rrreeepppooorrrttt uuunnnaaauuuttthhhooorrriiizzzeeeddd uuussseee tttooo cccuuusssttt---ssseeerrrvvv@@@aaammmsss...ooorrrggg... TTThhhaaannnkkk YYYooouuu!!! http://dx.doi.org/10.1090/stml/004 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 Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! This page intentionally left blank Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Problems in Mathematical Analysis I Real Numbers, Sequences and Series Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! This page intentionally left blank Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! 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 Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Editorial Board David Bressoud Carl Pomerance Robert Devaney, Chair Hung-Hsi Wu Originally published in Polish, as Zadania z Analizy Matematycznej. Cz$sc Pierwsza. Liczby Rzeczywiste, Ciagi i Szeregi Liczbowe © 1996, Wydawnictwo Uniwersytetu Marii Curie-Sklodowskiej, Lublin. Translated, revised and augmented by the authors. 2000 Mathematics Subject Classification. Primary 00A07, 40-01. Library of Congress Cataloging-in-Publication Data 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 98765432 09 Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! 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 Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! 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 Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Contents IX 3.7. Rearrangement of Series. Double Series 342 3.8. Infinite Products 360 Bibliography - Books 379 Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You!

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