Infinite Sequences and Series

Konrad Knopp

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486601533
• eBook ISBN: 9780486152042
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

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.

Chapter 1

INTRODUCTION AND PREREQUISITES

1.1. Preliminary remarks concerning sequences and series

In manifold investigations in pure and applied mathematics, it often happens that a result is not obtained all at once, as in 7 · 13 = 91, but that one seeks to approximate the result in a definite way by steps. This is the case, for example, in calculating the area of a circle of radius 1, for which we obtain first perhaps 3, then 22/7, say, then 3.1415, etc., as approximate values or approximations. This also occurs in the elementary method of calculating the square root of 2, where we get first 1, then 1.4, then 1.41, etc. We thus secure a sequence of values which lead, in a sense to be described later in more detail, to the value π,

, respectively. If we compute in such a manner, corresponding to each natural number 0, 1, 2, ...,¹ a number s₀, s₁, s₂, ..., or if these are given or defined in some other way, we say that we have before us an infinite sequence of numbers, or briefly a sequence, with the terms s_ν, and we denote it by

Such sequences may be given (generated, defined) in the most varied ways. Numerous examples will appear in the sequel. The following method occurs especially often: If a certain term of the sequence is already known, then the next term is given by means of the amount by which it differs from the former. E.g., if s₃ is already known, then s₄ is determined by indicating the difference s₄ − s₃ = a₄, so that s₄ = s₃ + a₄. Thus, a₄ represents the amount that has to be added to s₃ in order to obtain the next term s₄.³ If, for the sake of uniformity, we set the initial term s₀ equal to a₀, and then, in general, write s_ν − s_ν−1 = a_ν, we have, for n = 0, 1, 2, ...,

The n^th term of the sequence {s_ν} is obtained by “adding up” the terms of another (infinite) sequence {a_ν}. To indicate this continued summation process, the sequence thus obtained is denoted by

and is called an (infinite) series. The following is a simple example: If we divide out the fraction

according to the elementary rules, we get a sequence beginning with s₀ = 1, s₁ = 1 + a, s₂ = 1 + a + a² and yielding s_n−1 = 1 + a + ... + aⁿ⁻¹. The corresponding remainder

now yields, at the next step, aⁿ, which must be added to s_n−1 to produce s_n. Since this continues without end, we obtain the infinite series

Whether or not, and in what sense, this infinite series is the same as the fraction

which we started with, still requires, naturally, precise elucidation.

An (infinite) series is a means, employed particularly often in what follows, of defining an (infinite) sequence: A certain sequence {a_ν} is directly computed, defined–in short: given. It is, however, not itself the main object of the investigation; a new sequence {s_n} is derived from it, whose terms are formed according to the specification (2), and this sequence {s_n} is the one which furnishes the real subject of the investigation. Thus, in the series

we are not so much interested in the individual terms,

, of this series, as in what we get if we sum them up without end, i.e., form the sums

which approach the number 2 as the “value” of the infinite series. (For a precise definition, see 2.1.)

Of the two concepts sequence and series, the former is the simpler and more primitive one. In the first place, a series can only be defined if one already possesses the notion of a sequence; for to be able to write down the series (3), one must know the sequence of its terms. Furthermore, ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Foreword
• Chapter 1. Introduction and Prerequisites
• Chapter 2. Sequences and Series
• Chapter 3. The Main Tests for Infinite Series. Operating with Convergent Series
• Chapter 4. Power Series
• Chapter 5. Development of the Theory of Convergence.
• Chapter 6. Expansion of the Elementary Functions
• Chapter 7. Numerical and Closed Evaluation of Series
• Bibliography
• Index