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

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*W. J. Kaczor; M. T. Nowak*

We learn by doing. We learn mathematics by doing problems. This
book is the first volume of a series of books of problems in
mathematical analysis. It is mainly intended for students studying the
basic principles of analysis. However, given its organization, level,
and selection of problems, it would also be an ideal choice for
tutorial or problem-solving seminars, particularly those geared toward
the Putnam exam. The volume is also suitable for self-study.

Each section of the book begins with relatively simple exercises,
yet may also contain quite challenging problems. Very often several
consecutive exercises are concerned with different aspects of one
mathematical problem or theorem. This presentation of material is
designed to help student comprehension and to encourage them to ask
their own questions and to start research. The collection of problems
in the book is also intended to help teachers who wish to incorporate
the problems into lectures. Solutions for all the problems are
provided.

The book covers three topics: real numbers, sequences, and series,
and is divided into two parts: exercises and/or problems, and
solutions. Specific topics covered in this volume include the
following: basic properties of real numbers, continued fractions,
monotonic sequences, limits of sequences, Stolz's theorem, summation
of series, tests for convergence, double series, arrangement of
series, Cauchy product, and infinite products.

Also available from the AMS are Problems in Mathematical
Analysis II and Problems in Analysis
III in the Student Mathematical
Library series.

#### Readership

Undergraduates, graduate students, and instructors interested in analysis.

#### Reviews & Endorsements

A valuable resource.

-- American Mathematical Monthly

Would be an ideal choice for tutorial or problem-solving seminars. The volume is also suitable for self-study … presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research … a really useful book for practice in mathematical analysis.

-- Zentralblatt MATH

Belongs to the great tradition of Eastern European problem books … if you love mathematics and are serious about understanding analysis, this book is a must.

-- MAA Online

#### Table of Contents

# Table of Contents

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

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Notation and Terminology xiii14 free
- Problems 116 free
- Chapter 1. Real Numbers 318
- Chapter 2. Sequences of Real Numbers 1934
- Chapter 3. Series of Real Numbers 6378
- 3.1. Summation of Series 6378
- 3.2. Series of Nonnegative Terms 7287
- 3.3. The Integral Test 88103
- 3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz 92107
- 3.5. The Dirichlet and Abel Tests 99114
- 3.6. Cauchy Product of Infinite Series 102117
- 3.7. Rearrangement of Series. Double Series 105120
- 3.8. Infinite Products 112127

- Solutions 123138
- Chapter 1. Real Numbers 125140
- Chapter 2. Sequences of Real Numbers 151166
- Chapter 3. Series of Real Numbers 245260
- 3.1. Summation of Series 245260
- 3.2. Series of Nonnegative Terms 269284
- 3.3. The Integral Test 302317
- 3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz 309324
- 3.5. The Dirichlet and Abel Tests 324339
- 3.6. Cauchy Product of Infinite Series 333348
- 3.7. Rearrangement of Series. Double Series 342357
- 3.8. Infinite Products 360375

- Bibliography - Books 379394
- Back Cover Back Cover1396