Physical meaning of conditionally convergent series: the calculation of the Madelung constant

Abstract

Conditionally convergent series are infinite series whose result depends on the order of the sum. One of the most famous examples of conditionally convergent series of interest in Physics is the calculation of Madelung's constant α in ionic crystals. The appearance of this type of series is quite disturbing to students and often causes misunderstandings. In this work we analyze the physical meaning of the conditional convergence from a pedagogical point of view. The problem is posed using a toy model of ionic crystal in which the lattice sums can be calculated explicitly for various forms of expansion of the crystal about a central core. It is seen directly how the Coulomb series does not converge to α when there are charge accumulations on the surfaces. Therefore, it becomes clear what the appropriate strategy should be when choosing the order of summation to arrive at the correct value of α.