On quicker convergence towards Euler's constant.

In this paper, we deduced the following new asymptotic series Hn - ln n ~ γ + + 1/2(n + 1) + 5/12n(n+1) (1 + 1/5n + 1/50/n² - 1/50/n³ + 59/52500/n4 + 437/37500/n5 - ...) which faster converge to the Euler's constant with the increase in the terms considered, where Hn is the harmonic number. Also, we presented the following double inequality 1/2 + 5/12n(1+1/5/n+1/50/n²)/n+1 < Hn - ln n - γ < 1/2 + 5/12n(1+1/5/n)/n+1; n = 1, 2, 3..., which improved some known inequalities of the sequence Hn - ln n - γ. [ABSTRACT FROM AUTHOR]