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

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 $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \pm \ldots$