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We define a generalized Cesàro sequence spaceces(p), wherep=(pk)is a bounded sequence of positive real numbers, and consider it equipped with the Luxemburg norm. The main purpose of this paper is to show thatces(p)isk-nearly uniform convex (k-NUC) fork≥2whenlimn→∞infpn>1. Moreover, we also obtain that the Cesàro sequence spacecesp(where 1<p<∞)isk-NUC,kR, NUC, and has a drop property.doi:10.1155/s0161171203301267 fatcat:epz7d3cnmnbn5i3m3ekxuqbkpy