Terminal Summation: Extending the concept of convergence

Abstract

This paper presents an atypical method for summing divergent series, and provides a sum for the divergent series log(n). We use an idea of T.E. Phipps, called Terminal Summation, which uses asymptotic analysis to assign a value to divergent series. The method associates a series to an appropriate difference equations having boundary conditions at infinity, and solves the difference equations which then provide a value for the original series. We point out connections between Phipps' method, the Euler-MacLaurin sum formula, the Ramanujan sum and other traditional methods for summing divergent series.