Error estimates for the Gregory-Leibniz series and the alternating harmonic series using Dalzell integrals

TL;DR

This paper derives new error bounds for the Gregory-Leibniz and alternating harmonic series using Dalzell integrals, enhancing the understanding of their convergence properties.

Contribution

It introduces a novel approach employing Dalzell integrals to estimate errors in partial sums of these classical series.

Findings

New error estimates for the Gregory-Leibniz series

Improved bounds for the alternating harmonic series

Enhanced understanding of series convergence

Abstract

The computation of Dalzell integrals $\int_0^1 \frac{x^m (1-x)^n}{1+x^2} \, dx > 0$ gives new error estimates for the partial sums of the Gregory-Leibniz series $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} \pm \ldots$ and for the alternating harmonic series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \pm \ldots$