On Kummer's test of convergence and its relation to basic comparison tests

TL;DR

This paper explores Kummer's test for convergence of positive series, establishing a formal connection with basic comparison tests by constructing bounding series from the test sequence and vice versa.

Contribution

It provides a formal proof of the relationship between Kummer's test and comparison tests, including methods to construct bounding series from the test sequence.

Findings

Kummer's test encompasses Raabe's, Gauss's, and Bertrand's tests as special cases.

A formal method to transform the sequence {p_n} into a bounding series is established.

The paper offers a new approach to relate Kummer's test with basic comparison tests.

Abstract

Testing convergence of infinite series is an important part of mathematics. A very basic test of convergence is to upper-bound a given series with a known series, term by term. In $19^{th}$ century, Kummer proposed a test of convergence for any positive series based on finding a suitable positive sequence $\{p_n\}$ and a suitable real constant $c$. It can be easily shown that by choosing appropriate sequence $\{p_n\}$, the Kummer's test yields other tests like Raabe's, Gauss' or Bertrand's as its special cases. In 1995, Samelson noted that there is another interesting relation between Kummer's test and basic comparison tests, particularly, that one can transform the sequence $\{p_n\}$ into a convergent bounding series, and he sketched a simple proof of this statement. In this paper, we fill the missing formal proof, although using a different approach, and we show how to construct a bounding series from the sequence $\{p_n\}$ and vice versa.