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

Convergent series for lattice models with polynomial interactions