Summary: We introduce a new class of sequences of the form \[ \mu_n=\sum^n_{k=1} \frac 1k +\ln(e^{a/(n+b)}-1)-\ln a \] which converge to the Euler-Mascheroni constant \(\gamma\). Being preoccupied with the acceleration of the classical sequence convergence towards \(\gamma\), N. Batir [JIPAM, J. Inequal. Pure Appl. Math. 6, No. 4, Paper No. 103 (2005; Zbl 1089.33001)] and H. Alzer [Expo. Math. 24, No. 4, 385–388 (2006; Zbl 1105.11003)] studied the case \(a=b=1\); and we show in this paper that the fastest sequence \((\mu_n)_{n\geq 1}\) is obtained for \(a=1/\sqrt{2},\;b=(2+\sqrt{2})/4\). For these values, accurate approximations of \(\gamma\) can be constructed as numerical computations made in the final part of this paper show. We also solve an open problem about the rate of convergence of some sequences defined by Batir.