We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable domain, and show that its zero set actually coincides with the set of eigenvalues of J in that domain. Further we derive sufficient conditions under which the spectrum of J is approximated by spectra of truncated finite-dimensional Jacobi matrices. As an application we construct several examples of Jacobi matrices for which the characteristic function can be expressed in terms of special functions. In more detail we study the example where the diagonal sequence of J is linear while the neighboring parallels to the diagonal are constant.