Perturbation theory with convergent series for calculating physical quantities specified by finitely many terms of a divergent series in traditional perturbation theory

A new high-accuracy method is suggested for calculating physical quantities for which only finitely many terms of the divergent series in a traditional perturbation theory are known. The method is based on approximating the desired quantity with the sum of finitely many terms of an absolutely convergent series. As an example, the $\beta$-function in the $\varphi_4^4$ model and the critical exponent $\alpha$ characterizing the behavior of the $\mathrm{He}^4$ heat capacity near the phase transition point are calculated.