The buckling and post-buckling behavior of a nonlinear discrete repetitive system, the discrete \emph{elastica}, is studied herein. The nonlinearity essentially comes from the geometrical effect, whereas the constitutive law of each component is reduced to linear elasticity. The paper primarily focuses on the relevancy of higher-order continuum approximations of the difference equations, also called continualization of the lattice model. The pseudo-differential operator of the lattice equations are expanded by Taylor series, up to the second or the fourth-order, leading to an equivalent second-order or fourth-order gradient elasticity model. The accuracy of each of these models is compared to the initial lattice model and to some other approximation methods based on a rational expansion of the pseudo-differential operator. It is found, as anticipated, that the higher level of truncation is chosen, the better accuracy is obtained with respect to the lattice solution. This paper also outlines the key role played by the boundary conditions, which also need to be consistently continualized from their discrete expressions. It is concluded that higher-order gradient elasticity models can efficiently capture the scale effects of lattice models.