The best bounds of harmonic sequence

Abstract

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 constant. The constants $\frac{1}{1-\gamma }-2$ and $\frac{1}{3}$ are the best possible. As by-products, two double inequalities of the digamma and trigamma functions are established.

5 pages

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