We consider the sum ζ_N = ξ₁ + ... + ξ_N of independent identically distributed integer-valued random variables. The problem of estimating the probability P{ζ_N = n} as N →∞ in the case of large deviations was solved in Tkachuk’s theorem. We prove a similar theorem for the series scheme where the summand distribution depends on an unknown slowly varying function and has finite expectation and infinite variance.