Convergence of Madelung-like lattice sums

We make a general study of the convergence properties of lattice sums, involving potentials, of the form occurring in mathematical chemistry and physics. Many specific examples are studied in detail. The prototype is Madelung’s constant for NaCl: {equation*}∑ -∞∞ {(-1)ⁿ⁺ᵐ⁺ᵖ} {√n^2+m^2+p^2} = -1.74756459 ⋯ , {equation*} presuming that one appropriately interprets the summation proccess.