A New Differential Test for Series of Positive Terms

TL;DR

This paper introduces a novel differential test for determining the convergence or divergence of series with positive terms, based on the behavior of a specific function involving the reciprocal of the series terms.

Contribution

It presents a new, universal differential criterion for series of positive terms, expanding the tools available for convergence analysis.

Findings

The test effectively distinguishes convergent and divergent series.

The method is applicable to a wide class of positive-term series.

It simplifies convergence testing using differential properties.

Abstract

A new differential test for series of positive terms is proved. Let f(x) be a positive continuous function corresponded to a series of positive terms f(k), and g(x) is a derivative of reciprocal of f(x). Then, the convergence and divergence of the series may be determined from a value of fgx for enough large x. The rest may make the limit form, and is universal and complete.