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

396 Pages·2000·21.366 MB·English

by  W. J. Kaczor, M. T. Nowak

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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 [e-mail].
Thank You!
This page intentionally left blank
Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [e-mail].
Thank You!
Problems in
Mathematical
Analysis I
Real Numbers,
Sequences and Series
Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [e-mail].
Thank You!
This page intentionally left blank
Copyright 2000 American Mathematical Society. Duplication prohibited. Please report unauthorized use to [e-mail].
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 [e-mail].
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
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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
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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
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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 [e-mail].
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