Two accelerated approximations to the Euler-Mascheroni constant.

Let γ = 0.577215 ... be the Euler-Mascheroni constant, and let Pn = Σ kn = 1 1/k - 1/2 ln (n2 + n + 1/3) and Qn = Σ kn = 1 1/k - 1/4 ln [ (n2+n2+1/3)2-1/45] We prove that for all integers n ⩾ 1, 1/180(n+1)4 < γ - Pn < 1/180n4 and 8/2835(n+1)6 < Qn-γ < 8/2835n6. This provides the higher order estimate for the Euler-Mascheroni constant. [ABSTRACT FROM AUTHOR]