Convergence of a Family of Series

TL;DR

This paper introduces a family of expressions to analyze the convergence of related series, providing a systematic approach and applications for determining convergence based on new theorems.

Contribution

It develops a novel framework for assessing series convergence through a constructed family of expressions and offers practical applications of these theoretical results.

Findings

A new family of expressions for convergence analysis

Theorems linking expressions to series convergence

Applications demonstrating the method's utility

Abstract

In this article we construct a family of expressions $\varepsilon(n)$. For each element E(n) from $\varepsilon(n)$, the convergence of the series $\sum_{n \ge n_E}{E(n)}$ can be determined in accordance to the theorems of this article. Some applications are also presented.