Limits and inequalities associated with the Euler-Mascheroni constant

(i) We present several limits associated with the Euler-Mascheroni constant. (ii) Let γ=0.577215… be the Euler-Mascheroni constant, and let Tn=∑k=1n1k-lnn+12+124n and Pn=∑k=1n22k-1-ln(4n). We determine the best possible constants α,β,a and b such that the inequalities148(n+α)3⩽γ-Tn<148(n+β)3and124(n+a)2⩽Pn-γ<124(n+b)2are valid for all integers n⩾1.