# Analytic Summability Theory

• Ibrahim M. Alabdulmohsin
Chapter

## Abstract

The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose prototypical examples include the Abel summation method, the Cesaro means, and the Borel summability method. As will be established in subsequent chapters, the theory of summability of divergent series is intimately connected to the theory of fractional finite sums. In this chapter, we introduce a generalized definition of series as well as a new summability method for computing the value of series according to such a definition. We show that the proposed summability method is both regular and linear, and that it arises quite naturally in the study of local polynomial approximations of analytic functions. The materials presented in this chapter will be foundational to all subsequent chapters.

## Keywords

Summability Method Divergent Series Local Polynomial Approximation Euler-Maclaurin Summation Formula Mittag-Leffler Star
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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