# About some speeds of convergence to the constant of Euler

## DOI:

https://doi.org/10.31926/but.mif.2022.2.64.2.13## Keywords:

Constant of Euler, harmonic number, logarithm## Abstract

The speed of convergence of the classical sequence which defines the constant of Euler (or Euler-Mascheroni),* γ = lim _{n→∞} γ_{n} = 0*, 577215 . . . , where

*γ*, was intensively studied. In 1983 I established in [14] one of the first two sided estimates of this speed, namely 1/2n+1 < γ

_{n}=(∑_{k=1}^{n}1/k ) − ln n_{n}−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.