Floor function, integral, and others

The expression is equal to $\begin{aligned} \lim_{n\to\infty}\left(\sum_{k=1 , \Delta k=1}^n\frac{\Delta k}{k} -\int_1^n\frac{dx}{x}\right)&= \lim_{n\to \infty}(H_n - \ln n)\\&= \lim_{n\to\infty} \left(\gamma +\ln n+\frac{1}{2n}-O(n^{-2})-\ln n\right)=\gamma\end{aligned}$