Can someone help me prove that the limit approaches zero ? I know it does, but I can't prove it.
$$\lim_{n\to\infty}\sum\limits_{k=1}^n\frac{\ln k}{n}\bigg(1-\bigg\{\frac{n}{k}\bigg\}\bigg)\bigg(\frac{1}{2}-\frac{k}{n}\bigg\{\frac{n}{k}\bigg\}\bigg)$$
where $\displaystyle\left\{\frac{n}{k}\right\}$ is the fractional part of $\displaystyle \frac{n}{k}$.