Erdős problem 314

Let $n\geq 1$ and let $m$ be minimal such that $\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$. We define $$\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1.$$ How small can $\epsilon(n)$ be? Is it true that $$\liminf n^2\epsilon(n)=0?$$