Our goal is to find an asymptotic behavior as n→∞ of orthogonal polynomials P _n(z) defined by the Jacobi recurrence coefficients a _n,b _n. We suppose that the off-diagonal coefficients a _n grow so rapidly that the series ∑a _n ⁻¹ converges, that is, the Carleman condition is violated. With respect to diagonal coefficients b _n we assume that −b _n(a _na _n−1) ^−1/2→2β _∞ for some β _∞≠±1. The asymptotic formulas obtained for P _n(z) are quite different from the case ∑a _n ⁻¹=∞ when the Carleman condition is satisfied. In particular, if ∑a _n ⁻¹<∞, then the phase factors in these formulas do not depend on the spectral parameter z∈C. The asymptotic formulas obtained in the cases |β _∞|<1 and |β _∞|>1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.