A BBP-type formula for the remainder of the Madhava-Gregory-Leibniz series

TL;DR

This paper derives a BBP-type formula in base-16 for the remainder of the Madhava-Gregory-Leibniz series for pi, providing a new closed-form expression involving Pochhammer denominators, and also presents an analogous formula for log 2.

Contribution

It introduces a novel BBP-type formula for the series remainder of pi and log 2, expanding the toolkit for digit extraction and analysis of these constants.

Findings

Derived a closed-form BBP-type formula for pi's series remainder.

Presented an analogous BBP-type formula for log 2.

Provides new insights into the structure of these mathematical series.

Abstract

We derive a BBP-type formula for the remainder of the Madhava-Gregory-Leibniz series for $\pi$. The result is a closed form in base-$16$ with Pochhammer denominators. The analogous formula for the alternating series for $\log 2$ is also presented.