We show that defect modes in infinite systems of resonators have corresponding modes in finite
systems which converge as the size of the system increases. We study the generalized capacitance
matrix as a model for three-dimensional coupled resonators with long-range interactions and consider defect modes that are induced by compact perturbations. If such a mode exists, then there
are elements of the discrete spectrum of the corresponding truncated finite system that converge to
each element of the pure point spectrum. The rate of convergence depends on the dimension of the
lattice. When the dimension of the lattice is equal to that of the physical space, the convergence is
exponential. Conversely, when the dimension of the lattice is less than that of the physical space,
the convergence is only algebraic, because of long-range interactions arising due to coupling with
the far field.