The Summation of Series

The Summation of Series

by Harold T. Davis

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Overview

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.

Product Details

ISBN-13: 9780486789682
Publisher: Dover Publications
Publication date: 02/18/2015
Series: Dover Books on Mathematics Series
Pages: 160
Product dimensions: 5.50(w) x 8.50(h) x (d)

About the Author

Harold Thayer Davis (1892–1974) was Professor of Mathematics at Northwestern University. His other Dover book is Introduction to Nonlinear Differential and Integral Equations.

Table of Contents

Chapter 1 The Calculus of Finite Differences

1 Finite Differences 1

2 Factorial Symbols 2

3 Table of Differences and its Application 4

4 Differences of Higher Order 8

5 The Gregory-Newton Interpolation Formula 9

6 Summations 12

7 Operational Devices in the Calculus of Finite Differences 17

8 The Euler-Maclaurin Formula for Numerical Integration 20

9 Summary and Review 23

Chapter 2 The Gamma and Psi Functions

1 The Gamma Function 26

2 Properties of the Gamma Function 28

3 Generalizations of Factorial x 29

4 The Psi Function 31

5 Properties of the Psi Function 34

6 The Summation of Reciprocal Polynomials 35

7 Polygamma Functions 41

8 The Summation of Reciprocal Polynomials with Repeated Factors 44

Chapter 3 Other Methods of Summation

1 Summation by Differences 49

2 Logarithmic Numbers 52

3 Lubbock's Summation Formula 55

4 Lubbock's Summation Formula in Terms of Differences 57

5 Summation by Means of the Euler-Maclaurin Formula-The Bernoulli Polynomials 62

6 Euler's Constant and Other Limiting Values 66

7 Sums Involving Binomial Coefficients 69

8 Moments of the Bernoulli Distribution 74

Chapter 4 Summation By Tables

1 The Tabulation of Sums 77

2 Summation By Parts 77

3 Formulas Involving Sines and Cosines 79

4 Sums of Powers of Sines and Cosines 81

5 Some Miscellaneous Examples 84

6 The Summation of Logc Λ(x) 85

7 The Summation of xp log Λ(x) 88

8 The Summation of xp log x 89

Chapter 5 Infinite Sums

1 Infinite Sums 92

2 Tests for Convergence of Infinite Series 92

3 The Summation of Infinite Series 99

4 The Method of Taylor's Series 100

5 The Method of Inverse Differences 103

6 The Sums of Powers of Reciprocal Roots 106

7 Poisson's Formula 112

8 Comparison of the Euler-Maclaurin and Poisson Formulas 118

Table of Finite Sums

A General Forms 125

B Forms Involving Rational Coefficients and Powers of x 125

C Forms Involving ax and 2X 128

D Forms Involving ex 130

E Forms Involving Trigonometric Functions 131

F Forms Involving Logarithms 134

G Forms Involving Arc Tangents 134

H Miscellaneous Forms 135

Indexes

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