New Fast Convergent Sequences of Euler-mascheroni Type

This sequence has diverse applications in many areas of mathematics, ranging from classical or numerical analysis to number theory, special functions or theory of probability. There is a huge literature about the sequence (γn)n≥1 and the constant γ. Please refer to [3, 4, 5, 6, 7, 8] and all the references therein. The speed of convergence to γ of sequence (γn)n≥1 is very slowly, if we take into account that it converges like n−1. That is why many authors develop studies to improve the speed of convergence of sequence (γn)n≥1. For example, Cesaro [1] proved that for every positive integer n ≥ 1, there exists a number cn ∈ (0, 1) such that the following relation is true: