How to find convergence of conditionally convergent series obtained while calculating the electrostatic potential energy of a NaCl crystal? I was reading Electricity and Magnetism by E M Purcell and there in the first chapter there is an attempt to estimate the electrostatic potential energy of the crystal lattice of a NaCl crystal. 
During the calculation we obtain an infinite series which is conditionally convergent. Now, this reminded me of the Riemann Theorem regarding conditionally convergent sequences which says that a rearrangement of such a sequence can be made  to diverge or converge to any real number.
Now, we know that the order in which we are calculating the potential energy should not matter. But here, there is crude way described by the author to calculate the energy, and I'm unable to understand the exact reason behind the method employed.

It is clear, incidentally, that this series does not converge absolutely; if we were so foolish as to try to sum all the positive terms first, that sum would diverge.To evaluate such a sum, we should arrange it so that as we proceed
  outward, including ever more distant ions, we include them in groups
  that represent nearly neutral shells of material. Then if the sum is broken
  off, the more remote ions that have been neglected will be such an
  even mixture of positive and negative charges that we can be confident
  their contribution would have been small. This is a crude way to describe
  what is actually a somewhat more delicate computational problem. The
  numerical evaluation of such a series is easily accomplished with a computer.

I suspect that the method employed is justified in a sense that the assumption that it is an infinite series is wrong and hence the series will converge to a unique finite number. But, I'm not very sure and hence asking this question to know about others' views regarding this.
Thanks in advance.
 A: 
Now, this reminded me of the Riemann Theorem regarding conditionally convergent sequences which says that a rearrangement of such a sequence can be made to diverge or converge to any real number.

You are right. Just rearranging the order of summation on can get every real number or $ \pm \infty$ or even no limit at all. However it is false that the order in which we are calculating the potential energy should not matter. That is true for finite sums. Series are a different story. However series are unavoidable if one needs to say something about thermodynamic limit, i.e. if we need the energy per particle of the infinite system.
Although rarely cited, the real reason behind the conditional convergence of the Coulomb series is that for Coulomb point-like charges the boundary effects matter, they depend on the shape and charge of the surface  and do not vanish relatively to the bulk contribution with increasing the size of the sample, as it happens for short-range potentials. Therefore, there is a clear physical origin of the conditional convergence of the sum. The  only way to get a unique result is to decide which, among all possible boundary effects has to be chosen. It turns out that the choice of finite size cells which make vanishing the highest multipoles over the whole summation procedure  is a wise choice because it allows to get a unique result and energy per particle of different structures which are comparable without need of taking into account the pesence of surface terms.
Only after fixing the physical ambiguity of the problem, mathematics re-enter the game with special techniques (like Ewald summation or others) to speed up the convergence of a series made well behaving on the basis of a physical requirement. 
A couple of relevant references are:


*

*Roy, S. K. (1954). On the evaluation of certain lattice series. Canadian Journal of Physics, 32(8), 509-514.

*Hajj, F. Y. (1979). A simple method for the potential in ionic crystals. The Journal of Chemical Physics, 70(9), 4369-4375.

