A Short and Unified Proof of Kummer's Test

TL;DR

This paper provides a concise, unified proof of Kummer's test, a criterion for the convergence of positive series, enhancing understanding and application in mathematical analysis.

Contribution

The paper offers an exact analysis and a simplified proof of Kummer's test, clarifying its conditions and broadening its applicability.

Findings

Unified proof simplifies understanding of Kummer's test

Clarifies conditions for series convergence

Applicable to differential equations and mathematical philosophy

Abstract

Kummer's test from 1835 states that the positive series $\sum_{n=1}^\infty a_n$ is convergent if and only if there is a sequence $\{ B_n\}_1^\infty$ of positive numbers such that $B_n\cdot \frac{a_n }{a_{n+1}} -B_{n+1}\geq 1 ,$ for all sufficiently large $n$. We present an exact analysis and a short and unified proof of Kummer's test. The test has been applied to differential equations and studied in mathematical philosophy.