Let $n\ge 1$ and let $m$ be minimal such that ${\sum }_{n\le k\le m}\frac{1}{k}\ge 1$. We define$$ϵ(n)=\sum _{n\le k\le m}\frac{1}{k}-1.$$How small can $ϵ(n)$ be? Is it true that$$\mathrm{liminf}{n}^2ϵ(n)=0?$$