On quicker convergence towards Ruler's constant

In this paper, we deduced the following new asymptotic series H-n - ln n similar to gamma + 1/2(n + 1) + 5/12n(n + 1) (1 + 1/5/n + 1/50/n(2) - 1/50/n(3) + 59/52500/n(4) + 437/37500/n(5) - ...) which faster converge to the Euler's constant with the increase in the terms considered, where H-n is the harmonic number. Also, we presented the following double inequality 1/2 + 5/12n(1+1/5/n+1/50/n(2)) / n + 1 < H-n - ln n - gamma < 1/2 + 5/12n(1+1/5/n) / n + 1; n = 1, 2, 3, ... , which improved some known inequalities of the sequence H-n - ln n - gamma.