About one generalisation of the Leibniz theorem

TL;DR

This paper generalizes the Leibniz theorem for alternating series where the absolute terms are not necessarily monotonically convergent to zero, and discusses the accuracy of series remainder estimates.

Contribution

It introduces a broader generalization of the Leibniz theorem applicable to non-monotonically converging absolute terms in alternating series.

Findings

Generalization of Leibniz theorem for non-monotonically converging absolute terms

Analysis of accuracy in series remainder estimation

Extension of convergence criteria for alternating series

Abstract

The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonously" convergent to zero. Questions of accuracy of the estimation for the series remainder are considered.