A new summability method for divergent series

TL;DR

This paper introduces a new summability method for divergent series that is simple, regular, linear, and useful for practical computations, with theoretical analysis and applications in approximating analytic functions.

Contribution

It presents a novel summability method that is easy to implement, theoretically sound, and naturally arises in polynomial approximation of analytic functions.

Findings

The new method is regular and linear.

An asymptotic error expression is derived.

It is practical for computing divergent sums.

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 Borel summability method. In this paper, we introduce a new summability method for divergent series, and derive an asymptotic expression to its error term. We show that it is both regular and linear, and that it arises quite naturally in the study of local polynomial approximations of analytic functions. Because the proposed summability method is conceptually simple and can be implemented in a few lines of code, it can be quite useful in practice for computing the values of divergent sums.