Methods for the Summation of Series

Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Contents -- Foreword -- Testimonial -- Preface -- Biography -- Symbols -- 1. Classical Methods from Infinitesimal Calculus -- 1.1. Use of Infinitesimal Calculus -- 1.1.1. Convergence of series -- 1.1.2. Limits of sequences and series -- 1.2. Abel's Summation -- 1.2.1. Abel's theorem and Tauber theorem -- 1.2.2. Abel's summation method -- 1.3. Series Method -- 1.3.1. Use of the calculus of finite difference -- 1.3.2. Application of Euler-Maclaurin formula and the Bernoulli polynomials -- 2. Symbolic Methods -- 2.1. Symbolic Approach to Summation Formulas of Power Series -- 2.1.1. Wellknown symbolic expressions -- 2.1.2. Summation formulas related to the operator (1 - xE)-1 -- 2.1.3. Consequences and examples -- 2.1.4. Remainders of summation formulas -- 2.1.5. Q-analog of symbolic operators -- 2.2. Series Transformation -- 2.2.1. An extension of Eulerian fractions -- 2.2.2. Series-transformation formulas -- 2.2.3. Illustrative examples -- 2.3. Summation of Operators -- 2.3.1. Summation formulas involving operators -- 2.3.2. Some special convolved polynomial sums -- 2.3.3. Convolution of polynomials and two types of summations -- 2.3.4. Multifold Convolutions -- 2.3.5. Some operator summation formulas from multifold convolutions -- 3. Source Formulas for Symbolic Methods -- 3.1. An Application of Mullin-Rota's Theory of Binomial Enumeration -- 3.1.1. A substitution rule and its scope of applications -- 3.1.2. Various symbolic operational formulas -- 3.1.3. Some theorems on Convergence -- 3.1.4. Examples -- 3.2. On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas -- 3.2.1. A pair of (84) degree formulas -- 3.2.2. Specializations and examples -- 3.2.3. A further investigation of a source formula -- 3.2.4. Various consequences of the source formula.

Author: Tian-Xiao He

Publisher: Chapman & Hall/CRC

Publication Date: May 27, 2024

Number of Pages: NA pages

Language: English

Binding: Paperback

ISBN-10: 1032195002

ISBN-13: 9781032195001