Summability Calculus
Ibrahim M. Alabdulmohsin

TL;DR
Summability Calculus unifies and extends various mathematical results, allowing for direct analysis and manipulation of finite sums with periodic components, including differentiation, asymptotic analysis, and summation of divergent series.
Contribution
It introduces Summability Calculus, a new framework that simplifies and generalizes the analysis of finite sums, encompassing many classical theorems and identities in a unified manner.
Findings
Provides a method to differentiate and integrate finite sums with respect to bounds.
Extends classical results like Euler-Maclaurin and Stirling's approximation.
Derives numerous historical identities using elementary rules of Summability Calculus.
Abstract
In this paper, we present the foundations of Summability Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite differences under a single simple umbrella. Using Summability Calculus, any given finite sum of the form , where is an arbitrary periodic sequence, becomes immediately \emph{in analytic form}. Not only can we differentiate and integrate with respect to the bound without having to rely on an explicit analytic formula for the finite sum, but we can also deduce asymptotic expansions, accelerate convergence, assign natural values to divergent sums, and evaluate the finite sum for any . This follows because the discrete definition of the simple finite sum embodies…
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Taxonomy
TopicsApproximation Theory and Sequence Spaces · Advanced Mathematical Identities · Mathematical functions and polynomials
