Abstract
An efficient and simple algorithm, “improved Euler's transformation of series”, for accelerating the rate of convergence of an alternating series which only needs the convolved binomial coefficients as the damping factor, is presented. In order to apply this algorithm to the problem of the lattice sum for the electrostatic energy of ionic crystals, several model lattices are introduced, i.e., centroid lattice, mean lattice, binomial lattice, Evjen lattice, and Euler lattice. The improved Euler's transformation is extended to the problem of obtaining the Madelung constants of up to 4-dimensional cubic lattices by the use of multilayers of fractional dummy points. It was found that this method gives accurate series limits with a very small number of terms even for the cases where the conventional Evjen's method fails to be applicable. Since Evjen's method is shown to be a special case of Euler's transformation, the proposed method is interpreted as an extended Evjen's method.
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