The best bounds of harmonic sequence

Abstract

For any natural number $n\in\mathbb{N}$, $ \frac{1}{2n+\frac1{1-γ}-2}\le \sum_{i=1}^n\frac1i-\ln n-γ<\frac{1}{2n+\frac13}, $ where $γ=0.57721566490153286...m$ denotes Euler&#39;s constant. The constants $\frac{1}{1-γ}-2$ and $\frac13$ are the best possible. As by-products, two double inequalities of the digamma and trigamma functions are established.