The best bounds of harmonic sequence

Chao-Ping Chen, Feng Qi

For any natural number $n\in \mathbb{N}$, $\frac{1}{2n+\frac{1}{1-\gamma }-2}\le {\sum }_{i=1}^n\frac{1}{i}-\ln n-\gamma <\frac{1}{2n+\frac{1}{3}},$ where $\gamma =\mathrm{0.57721566490153286...}m$ denotes Euler's const…