About some speeds of convergence to the constant of Euler

The speed of convergence of the classical sequence which defines the constant of Euler (or Euler-Mascheroni), γ = lim n→∞ γn = 0, 577215 . . . , where γn =(∑k=1n 1/k ) − ln n, was intensively studied. In 1983 I established in [14] one of the first two sided estimates of this speed, namely 1/2n+1 < γn−γ < 1/2n. Further several new sequences with a faster convergence are defined either by modifying the argument of the logarithm (De Temple, 1993, Negoi 1997, Ivan 2002) or by modifying the last term 1/n of the harmonic sum (Vernescu 1999). Now we give a systematic study of these speeds of convergence and especially of the last ones.