The best bounds of harmonic sequence

Abstract

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