Infinite Sequences and Series - Brossura

Knopp, Konrad

Brossura

ISBN 10: 0486601536 ISBN 13: 9780486601533

Casa editrice: Dover Pubns, 1956

Riassunto

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.

Su questo libro

Contenuti:

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 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

Product Description:

Careful presentation of fundamentals of the theory by one of the finest modern expositors of higher mathematics. Covers functions of real and complex variables, arbitrary and null sequences, convergence and divergence, Cauchy's limit theorem, tests for infinite series, power series, numerical and closed evaluation of series.

Su questa edizione

• Editore: Dover Pubns
• Data di pubblicazione: 1956
• ISBN 10: 0486601536
• ISBN 13: 9780486601533
• Rilegatura: Copertina flessibile
• Numero di pagine: 208