Interpreting divergences and Madelung-type sums via extrapolations of sequence generating polynomials: Method of intersection and area rule

Abstract

Based on explorations in estimating certain Madelung constants, we put forward here two separate strategies to understand the meaning of two distinct classes of divergent non-power-series expansions. One class refers to alternating series representations, the other to monotonic ones. They chiefly rest on precise and approximate polynomial extrapolations, depending on situations. In case of sawtooth sequences, e.g., the partial-sums obtainable from Dirichlet eta or beta function at negative integer arguments, exact sequence-generating polynomials are found. Extrapolations yield a graphical meaning to anti-limit here, along with the exact answer. For staircase sequences, like the ones obtained from partial-sums of series representations for lambda and zeta functions, again at negative integer arguments, anti-limits do not exist. But, correct sequence-generating polynomials are obtained. There, our recipe relies on estimation of specific, finite areas embedded by such polynomials. The schemes put forward here are direct, independent and conceptually appealing. A subsequent extension of the latter strategy to alternating series also lends extra credence. Two new interpretations of summability are gained. Pilot calculations on several types of lattice sums reveal the worth of our endeavor with approximate extrapolations as well.