# $\lim_{n\to\infty} \sum\limits_{k=1}^n \frac{\ln k}{n} ( 1-\{\frac{n}{k} \} ) ( (1-\frac{k}{n} \{\frac{n}{k}\} )-\frac{1}{2})$

Can someone help me prove that the limit approaches zero, I know it does, but I can't prove it.

$$\lim_{n\to\infty} \sum\limits_{k=1}^n \frac{\ln k}{n} \left( 1-\left\{\frac{n}{k} \right\} \right) \left( \left(1-\frac{k}{n} \left\{\frac{n}{k}\right\} \right)-\frac{1}{2} \right)$$

where $\displaystyle \left\{ \frac{n}{k} \right\}$ is the fractional part of $\displaystyle \frac{n}{k}$.

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Please consider adding $\LaTeX$ format to your question –  Pragabhava Oct 26 '12 at 2:24
how do i do that –  boby Oct 26 '12 at 2:26
@boby I typeset your fomulae using $\LaTeX$, please check if my interpretation is correct. Also to other members, is there a better notation for the Fractional function? –  FrenzY DT. Oct 26 '12 at 2:28
Thanks a bunch, could you help me prove it though? I tested it on wolfram alpha and it aproaches zero quite fast but I still cant prove it –  boby Oct 26 '12 at 2:29
To answer your question "how do i do that": meta.math.stackexchange.com/questions/107/… –  joriki Oct 26 '12 at 4:18