Theory of Normalized Remainders in Taylor Series Expansions

TL;DR

This paper explores the concept of normalized remainders in Taylor series, detailing its historical background, mathematical significance, and recent formalization within the broader dynamical framework.

Contribution

It introduces and formalizes the normalized remainder concept, embedding it into a deeper mathematical and dynamical context, advancing the theoretical understanding.

Findings

Identified recurring patterns in normalized remainders

Developed a formal mathematical framework for normalized remainders

Embedded the concept within broader dynamical systems theory

Abstract

Since 2023, through the detailed examination of numerous concrete examples, the author and his collaborators have identified a recurring pattern. Building upon this observation, they introduced the concept of the normalized remainder. They deliberately chose this term and subsequently explored its historical background and mathematical significance. In 2026, Abu-Ghuwaleh propelled the subject forward at a deeper structural level. By exploring the broader dynamical and theoretical framework surrounding the normalized remainder family, he significantly developed and formalized the concept, firmly embedding it within the field. Consequently, the notion of the normalized remainder now carries richer and more profound mathematical significance. In this chapter, the author presents a synthesis of the research process and the principal findings related to the normalized remainder.