The Summation of Series

Harold T. Davis

• Publisher: Dover Publications
• Year: 2014
• Topic: Mathematics
• eBook ISBN:
  9780486801346
• Subtopic: Calculus
• Index: Mathematics

Valuable as both a text and a reference, this concise monograph starts with a consideration of the calculus of finite differences and advances to discussions of the gamma and psi functions and other methods of summation. Subsequent chapters offer a summation of tables and an examination of infinite sums. The treatment concludes with a table of finite sums and helpful indexes.
This volume was written by a prominent mathematician and educator whose interests encompassed the history of mathematics, statistics, modeling in economics, mathematical physics, and other disciplines. The book is suitable for students, researchers, and applied mathematicians in many areas of mathematics, computer science, and engineering.

CHAPTER 1

THE CALCULUS OF FINITE DIFFERENCES

1. Finite Differences. The demands not only of statistics, but of physical science as well, have made desirable some knowledge of what is called the calculus of finite differences to distinguish it from the infinitesimal calculus. The calculus of finite differences can be divided into two parts, one concerned with differences, called the difference calculus, and the other concerned with summations, called the summation calculus. These resemble in many details the differential and integral calculus.

Since the elements of the calculus of finite differences are readily understood by any one who has mastered the principles of the infinitesimal calculus, it will be possible to develop readily many of the important features of this very useful discipline.

By the difference of a function f(x), designated by Δf(x), or simply Δf, we mean the expression

where d is the difference interval.

That the differential calculus is in a sense the limiting form of the difference calculus is at once observed from the following limit:

In most of the discussion which follows it will simplify the analysis to assume that the difference interval is 1, that is to say, we shall concern ourselves with the special difference defined by

Examples of differences are given by the following:

The difference calculus can now be developed...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• Chapter 1. The Calculus of Finite Differences
• Chapter 2. The Gamma and Psi Functions
• Chapter 3. Other Methods of Summation
• Chapter 4. Summation by Tables
• Chapter 5. Infinite Sums
• Table Of Finite Sums
• INDEXES