We define a generalized Cesàro sequence space ces ( p ) , where p = ( p k ) 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 that ces ( p ) is k -nearly uniform convex ( k -NUC) for k ≥ 2 when 1$"> lim n → ∞ inf p n > 1 . Moreover, we also obtain that the Cesàro sequence space ces p ( where   1 < p < ∞ ) is k -NUC, k R , NUC, and has a drop property.