In this paper, we study the ideals of finite elements in special vector lattices of real sequences, first in the duals of Cesàro sequence spaces ces _p for p∈ { 0 } ∪ [1 , ∞) and, second, after the Cesàro sum ces _p(X) of a sequence of Banach spaces is introduced, where p= ∞ is also allowed, we characterize their duals and the finite elements in these sums if the summed up spaces are Banach lattices. This is done by means of a remarkable extension of the corresponding result for direct sums.