Borel summation of divergent series in field theory and Wynn's $\varepsilon$ algorithm

The first confluent form of Wynn's $\varepsilon$ algorithm is used in the Borel summation of some divergent perturbation-theory series that satisfy a strong asymptotic condition. The summation procedure reduces to the calculation of a sequence of ratios of Hankel functional determinants composed of a Borel integral and its derivatives and can be regarded as an alternative to the Padé and Padé–Borel methods. It admits a simple generalization to the summation of multiple series. The perturbation series for the ground-state energy of the anharmonic oscillator, Yukawa potential, and charmonium potential are analyzed; the critical exponents of the $O(n)$-symmetric $\varphi^4$ theories (models of phase transitions) for $n=0,1,2,3$ and the dilute Ising model are determined.