  The standard perturbative weak-coupling expansions in lattice models are
asymptotic. The reason for this is hidden in the incorrect interchange of the
summation and integration. However, substituting the Gaussian initial
approximation of the perturbative expansions by a certain interacting model or
regularizing original lattice integrals, one can construct desired convergent
series. In this paper we develop methods, which are based on the joint and
separate utilization of the regularization and new initial approximation. We
prove, that the convergent series exist and can be expressed as the re-summed
standard perturbation theory for any model on the finite lattice with the
polynomial interaction of even degree. We discuss properties of such series and
make them applicable to practical computations. The workability of the methods
is demonstrated on the example of the lattice $\phi^4$-model. We calculate the
operator $\langle\phi_n^2\rangle$ using the convergent series, the comparison
of the results with the Borel re-summation and Monte Carlo simulations shows a
good agreement between all these methods.
