Infinite Sequences and Series (eBook)

ForewordChapter 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 variableChapter 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 seriesChapter 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 productsChapter 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 seriesChapter 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 seriesChapter 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 functionsChapter 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