Sharp inequalities for the harmonic numbers (Q860367)

Let \(H_n=\sum_{k=1}^n1/k\) be the \(n\)-th harmonic number. The author proves that for all \(n\geq 1\) it holds \[ a-\log(e^{\frac{1}{n+1}}-1)\leq H_n<b-\log(e^{\frac{1}{n+1}}-1), \] with the best possible constants \(a=1+\log(\sqrt{e}-1)\) and \(b=\gamma\) -- Euler's constant.