Summation of series 级数和

This second, greatly enlarged edition of a work that has found favor among all who deal with series and their summations features more than 1,100 common series, all grouped for easy reference. Mathematicians, physicists, computer technicians, engineers, and students will find this volume immensely useful.
Arranged by category, these series include arithmetical and geomet-rical progressions, powers and products of natural numbers, figurate and polygonal numbers, inverse natural numbers, exponential and logarithmic series, binomials, simple inverse products, factorials, and trigonometrical and hyperbolic expansions. Additional series include those of various Bessel functions, elliptic integrals, and hypergeo-metric functions. A chapter is devoted to the relations between infi-nite products and infinite series, and other chapters discuss special series involving Legendre polynomials, the zeta functions, Bernoulli's functions, and similar expressions. The book concludes with a num-ber of rules for summing important, commonly encountered forms of series with no special names.
Each of the series appears not only with its summation but also with a bibliographical reference for more detailed information about the method of summing, applications, and proof. The range of the book can be further extended by differentiation and integration of the given series.

I. ARITHMETICAL PROGRESSION
II. GEOMETRICAL PROGRESSION
III. ARITHMETICAL AND GEOMETRICAL PROGRESSION
IV. POWERS OF NATURAL NUMBERS
V. PRODUCTS OF NATURAL NUMBERS
VI. FIGURATE AND POLYGONAL NUMBERS
VII. INVERSE NATURAL NUMBERS
Vlll. EXPONENTIAL AND LOGARITHMIC SERIES
IX. BINOMIALS
X. SIMPLE INVERSE PRODUCTS
XI. OTHER INVERSE PRODUCTS
XII. SIMPLE FACTORIALS
xIII. OTHER POWER SERIES (Bernoulli's and Euler's numbers)
IV. TRIGONOMETRICAL SUMMATIONS
XV. HYPERBOLIC SUMMATIONS
XVI. TRIGONOMETRICAL EXPANSIONS
XVll. HYPERBOLIC EXPANSIONS
XVIII. TAYLOR'S AND MACLAURIN'S THEOREM
XlX. BESSEL FUNCTIONS
XX. ELLIPTIC FUNCTIONS.
XXI. VARIOUS INTEGRALS.
XXII. BETA AND GAMMA FUNCTIONS
XXIlI. INFINITE PRODUCTS
XXIV. FOURIER'S SERIES
XXV. HYPERGEOMETRIC FUNCTIONS
XXVI. RELATIONS BETWEEN PRODUCTS AND SERIES
XXVII. SPECIAL FUNCTIONS
XXVIII. ZETA FUNCTIONS
XXIX. LEGENDRE POLYNOMIALS
XXX. SPECIAL PRODUCTS
XXXI. GENERAL FORMS
XXXII. DOUBLE AND TREBLE SERIES
XXXIlI. BERNOULLI'S FUNCTIONS
Bernoulli's Numbers
Table of Bernoulli's Numbers in Vulgar Fractions
Table of Bernoulli's Numbers in Integers and Repeating Decimals
Values of Constants in Series (305) to (318) and (1130)
Euler's Numbers
Euler's Constant
Sum of Power Series
Relations between Bernoulli's N umbers