An abstract description of the Richardson–Kalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem ${{\Rho}}$ is considered, the solution of which results in a real number $u$. To solve this problem, a numerical method is used, that is, the set ${H\subset \mathbb{R}}$ and the mapping ${u_h:H\to\mathbb{R}}$ are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set $H$ and ${u_h}$ can be expanded in a convergent series in powers of ${h:u_h=u+c_1h^k+...}$. In this very general situation, the Richardson–Kalitkin method is formulated for obtaining estimates for $u$ and $c$ from two values of ${u_h}$. The question of using a larger number of ${u_h}$ values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the Richardson–Kalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.