A Dilution Test for the Convergence of Subseries of a Monotone Series

TL;DR

This paper introduces a new convergence test that determines the convergence of any subseries of a monotone series by analyzing a weighted version of the original series, extending classical results like Cauchy's condensation test.

Contribution

It provides a novel method to assess subseries convergence of monotone series using a specially designed weighted series, establishing a converse to Cauchy's test.

Findings

New convergence test for subseries of monotone series

Extension of Cauchy's condensation test to arbitrary subseries

Proof of the equivalence between subseries convergence and weighted series convergence

Abstract

Cauchy's condensation test allows to determine the convergence of a monotone series by looking at a weighted subseries that only involves terms of the original series indexed by the powers of two. It is natural to ask whether the converse is also true: Is it possible to determine the convergence of an arbitrary subseries of a monotone series by looking at a suitably weighted version of the original series? In this note we show that the answer is affirmative and introduce a new convergence test particularly designed for this purpose.