  The band structures describing non-interacting particles in one-dimensional
superlattices of arbitrary periodicity are obtained by an analytical
diagonalization of the Hamiltonian without adopting the popular tight-binding
approximation. The results are compared with those of the tight-binding
approximation. In this way, a quantitative prediction of the validity and
failure of the tight-binding approximation becomes possible. In particular, it
is demonstrated that in contrast to the prediction of the tight-binding
approximation the central energy bands do not touch for periodicities $\tau$ of
the lattice where $\tau=4n$ and $n$ is an integer.
