Real Infinite Series

This is a widely accessible introductory treatment of infinite series of real numbers, bringing the reader from basic definitions and tests to advanced results. An up-to-date presentation is given, making infinite series accessible, interesting, and useful to a wide audience, including students, teachers, and researchers. Included are elementary and advanced tests for convergence or divergence, the harmonic series, the alternating harmonic series, and closely related results. One chapter offers 107 concise, crisp, surprising results about infinite series. Another gives problems on infinite series, and solutions, which have appeared on the annual William Lowell Putnam Mathematical Competition. The lighter side of infinite series is treated in the concluding chapter where three puzzles, eighteen visuals, and several fallacious proofs are made available. Three appendices provide a listing of true or false statements, answers to why the harmonic series is so named, and an extensive list of published works on infinite series.

This concise text focuses on the convergence of real series. Topics include functions and limits, real sequences and series, series of non-negative terms, general series, series of functions, the multiplication of series, more. 1959 edition.

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, more.

This book is a complete and self contained presentation on the fundamentals of Infinite Series and Products and has been designed to be an excellent supplementary textbook for University and College students in all areas of Math, Physics and Engineering.Infinite Series and Products is a branch of Applied Mathematics with an enormous range of applications in various areas of Applied Sciences and Engineering.The Theory of Infinite Series and Products relies heavily on the Theory of Infinite Sequences and therefore the reader of this text is urged to refresh his/her background on Sequences and related topics.In our e-book "Sequences of Real and Complex Numbers" the reader will find an excellent introduction to the subject that will help him/her to follow readily the matter developed in the current text.The content of this book is divided into 11 chapters.In Chapter 1 we introduce the Σ and the Π notation which is widely used to denote infinite series and infinite products, respectively. In Chapter 2 we present some basic, fundamental concepts and definitions pertaining to infinite series, such as convergent series, divergent series, the infinite geometric series, etc.In Chapter 3 we introduce the extremely important concept of Telescoping Series and show how this concept is used in order to find the sum of an infinite series in closed form (when possible). In this chapter we also present a list of Telescoping Trigonometric Series, which arise often on various applications.In Chapter 4 we develop some general Theorems on Infinite Series, for example deleting or inserting or grouping terms in a series, the Cauchy's necessary and sufficient condition for convergence, the widely used necessary test for convergence, the Harmonic Series, etc.In Chapter 5 we study the Convergence Test for Series with Positive Terms, i.e. the Comparison Test, the Limit Comparison Test, the D' Alembert's Test, the Cauchy's n-th Root Test, the Raabe's Test, the extremely important Cauchy's Integral Test, the Cauchy's Condensation Test etc.In Chapter 6 we study the Alternating Series and the investigation of such series with the aid of the Leibnitz's Theorem.In Chapter 7 we introduce and investigate the Absolutely Convergent Series and the Conditionally Convergent Series, state some Theorems on Absolute and Conditional Convergence and define the Cauchy Product of two absolutely convergent series.In Chapter 8 we give a brief review of Complex Numbers and Hyperbolic Functions, needed for the development of series from real to complex numbers. We define the Complex Numbers and their Algebraic Operations and give the three representations i.e. the Cartesian, the Polar and the Exponential representation of the Complex Numbers. The famous Euler's Formulas and the important De Moivre's Theorem are presented and various interesting applications are given. In this chapter we also define the so called Hyperbolic Functions of real and complex arguments.In Chapter 9 we introduce the theory of Series with Complex Terms, define the convergence in the complex plane and present a few important Theorems which are particularly useful for the investigation of series with complex terms.In Chapter 10 we define the Multiple Series and show how to treat simple cases of such series.In Chapter 11 we present the fundamentals of the Infinite Products, give the necessary and sufficient condition for the convergence of Infinite Products and define the Absolute and Conditional Convergence of Products. In particular in this chapter we present the Euler's product formula for the sine function and show how Euler used this product to solve the famous Basel problem.The 63 illustrative examples and the 176 characteristic problems are designed to help students sharpen their analytical skills on the subject.

"Higher mathematics" once pointed towards the involvement of infinity. This we label analysis. The ancient Greeks had helped it to a first high point when they mastered the infinite. The book traces the history of analysis along the risky route of serial procedures through antiquity. It took quite long for this type of mathematics to revive in our region. When and where it did, infinite series proved the driving force. Not until a good two millennia had gone by, would analysis head towards Greek rigor again. To follow all that trial, error and final accomplishment, is more than studying history: It provides touching, worthwhile access to advanced calculus. Moreover, some steps beyond convergence show infinite series to naturally fit a wider frame.

This text gives a rigorous treatment of the foundations of calculus. In contrast to more traditional approaches, infinite sequences and series are placed at the forefront. The approach taken has not only the merit of simplicity, but students are well placed to understand and appreciate more sophisticated concepts in advanced mathematics. The authors mitigate potential difficulties in mastering the material by motivating definitions, results and proofs. Simple examples are provided to illustrate new material and exercises are included at the end of most sections. Noteworthy topics include: an extensive discussion of convergence tests for infinite series, Wallis’s formula and Stirling’s formula, proofs of the irrationality of π and e and a treatment of Newton’s method as a special instance of finding fixed points of iterated functions.

This textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series. The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required. The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading. The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.