The Best Lower and Upper Bounds of Harmonic Sequence

For any natural number n ∈ N, 1 2n+ 1 1−γ − 2 ≤ n ∑ i=1 1 i − lnn− γ < 1 2n+ 1 3 , (1) where γ = 0.57721566490153286 · · · denotes Euler’s constant. The constants 1 1−γ − 2 and 1 3 are the best possible.