  Working at the Lifshitz level, we investigate the van der Waals interactions
across a series of layers with a periodic motif. We derive the complete form of
the van der Waals interaction as an explicit function of the number of periodic
layers. We then compare our result with an approximation based on an
anisotropic-continuum representation of the stratified medium. Satisfactory
agreement between discrete-layer and continuum models is reached only for
thicknesses of ten or more layers.
