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The Summation of Series

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Author(s): Davis, Harold T.
Publisher: Principia Press
Year: 1962

Language: English

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 a: 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 Loge T(x) 85
7. The Summation of xr log T(x) 88
8. The Summation of x* 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 and 2’ 128
D. Forms Involving e1 ................ 130
E. Forms Involving Trigonometric Functions 131
F. Forms Involving Logarithms 134
G. Forms Involving Arc Tangents 134
H. Miscellaneous Forms 135
INDEXES