Convex combinations of some convergent sequences

Abstract

We consider the convex combinations (Formula presented.), of a pair of sequences of real numbers (Formula presented.) and (Formula presented.) such that (Formula presented.), converging to (Formula presented.), and study the location of the limit inside the intervals (Formula presented.), for every (Formula presented.) or for sufficiently large (Formula presented.). We also investigate the same problem for the case of two corresponding sequences converging to (Formula presented.). Among other results, we prove some, a bit, unexpected ones. Namely, for each (Formula presented.), we determine the exact index (Formula presented.) at which the sequence (Formula presented.) changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented.