New series representation for Madelung constant

TL;DR

This paper introduces a novel series representation for the Madelung constant that converges rapidly, allowing accurate approximation even when the series is omitted, and presents additional related identities.

Contribution

It provides the fastest known series representation for the Madelung constant, combining an exact term with a rapidly converging series and deriving new identities.

Findings

Series converges exponentially fast

Approximate Madelung constant accurate to ten decimal places without the series

Introduces new mathematical identities related to Madelung constant

Abstract

A new series representation of the Madelung constant is given. We represent Madelung constant as a sum of an exact term plus an exponentially fast converging series. The remarkable result is that even if the series part is discarded, one obtains Madelung constant correct up to ten good decimal figures. This, to the best of our knowledge, may be the fastest converging series representation of the Madelung constant. A few other important identities are also obtained.