One of the finest expositors in the field of modern mathematics, Dr. Konrad Knopp here concentrates on a topic that is of particular interest to 20th-century mathematicians and students. He develops the theory of infinite sequences and series from its beginnings to a point where the reader will be in a position to investigate more advanced stages on his own. The foundations of the theory are therefore presented with special care, while the developmental aspects are limited by the scope and purpose of the book.
All definitions are clearly stated; all theorems are proved with enough detail to make them readily comprehensible. The author begins with the construction of the system of real and complex numbers, covering such fundamental concepts as sets of numbers and functions of real and complex variables. In the treatment of sequences and series that follows, he covers arbitrary and null sequences; sequences and sets of numbers; convergence and divergence; Cauchy's limit theorem; main tests for sequences; and infinite series. Chapter three deals with main tests for infinite series and operating with convergent series. Chapters four and five explain power series and the development of the theory of convergence, while chapter six treats expansion of the elementary functions. The book concludes with a discussion of numerical and closed evaluation of series.
Author(s): Konrad Knopp
Series: Dover Books on Mathematics
Edition: 1
Publisher: Dover Publications
Year: 1956
Language: English
Commentary: Covers, bookmarks, OCR, paginated.
Pages: 198
Tags: Математика;Математический анализ;Ряды;
Foreword
Chapter 1. INTRODUCTION AND PREREQUISITES
1.1. Preliminary remarks concerning sequences and series
1.2. Real and complex numbers
1.3. Sets of numbers
1.4. Functions of a real and of a complex variable
Chapter 2. SEQUENCES AND SERIES
2.1. Arbitrary sequences. Null sequences
2.2. Sequences and sets of numbers
2.3. Convergence and divergence
2.4. Cauchy's limit theorem and its generalizations
2.5. The main tests for sequences
2.6. Infinite series
Chapter 3. THE MAIN TESTS FOR INFINITE SERIES. OPERATING WITH CONVERGENT SERIES
3.1. Series of positive terms: The first main test and the comparison tests of the first and second kind
3.2. The radical test and the ratio test
3.3. Series of positive, monotonically decreasing terms
3.4. The second main test
3.5. Absolute convergence
3.6. Operating with convergent series
3.7. Infinite products
Chapter 4. POWER SERIES
4.1. The circle of convergence
4.2. The functions represented by power series
4.3. Operating with power series. Expansion of composite functions
4.4. The inversion of a power series
Chapter 5. DEVELOPMENT OF THE THEORY OF CONVERGENCE
5.1. The theorems of Abel, Dini, and Pringsheim
5.2. Scales of convergence tests
5.3. Abel's partial summation. Lemmas
5.4. Special comparison tests of the second kind 132 ·
5.5. Abel's and Dirichlet's tests and their generalizations
5.6. Series transformations
5.7. Multiplication of series
Chapter 6. EXPANSION OF THE ELEMENTARY FUNCTIONS
6.1. List of the elementary functions
6.2. The rational functions
6.3. The exponential function and the circular functions
6.4. The logarithmic function
6.5. The general power and the binomial series
6.6. The cyclometric functions
Chapter 7. NUMERICAL AND CLOSED EVALUATION OF SERIES
7.1. Statement of the problem
7.2. Numerical evaluations and estimations of remainders
7.3. Closed evaluations
BIBLIOGRAPHY
INDEX
