On Selecting an Optimal α in summing α-convex Series

Abstract

Summing Leibnitz alternating series using -convexity conditions imposed on the terms of theseries is a new concept. The art of choosing an optimal  for accelerating the summation procedure, based on these -onvexity conditions is discussed and analyzed in the existingliterature. For a particular class of series this choice of an optimal  readily gives the exactsum and this is analyzed in the light of optimization theory. The aim of this note is to relatethe concept of choosing the optimal  for summing this particular class of alternating series to that of the linear programming techniques. The -convexity imposed on the terms of theseries are exploited which determine a range of  and this range also is the convex region fordetermining the optimal . For the sake of clarity, several numerical examples are worked out.