Generalization of Jamet's test for convergence of number series and its new modifications

TL;DR

The paper addresses convergence criteria for positive number series by generalizing and unifying Jamet's test with logarithmic and Schlomilch-type criteria. It introduces a hierarchical framework built on the iterated logarithm sequence $\lambda_0(n)=1$, $\lambda_1(n)=n$, $\lambda_{k+1}(n)=\log(\lambda_k(n))$, along with auxiliary scaling functions $f,g,\varphi,\zeta$ to derive generalized tests such as $L_{k,n}$, $S_{k,n}$, and $\Omega_{k,n}$. The results establish generalized Jamet-type criteria for finite and infinite limits of a controlling parameter $q$ (including $q\to+\infty$ and $q=0$) and introduce a broad family of Schlomilch-based modification tests, with corresponding Theorems 3–5 and their variants. A second part extends these ideas via the Phi-class $\Phi$, yielding Theorems 6–10 and variants that cover a wide range of comparator functions and reproduce classic results for simple choices, demonstrating broad applicability to establishing convergence/divergence of number series.

Abstract

In this article, we present new generalizations of logarithmic convergence tests for number series, from which we will derive various new generalizations of the Jamet's convergence test. Further, similarly, on the basis of the generalizations of the Schlomilch's test we found, we will obtain modified tests of the convergence of number series.