Find

$\newcommand{\pars}[1]{\left\{ \frac{n}{#1} \right\}}$

$$\lim_{n\to\infty}\dfrac{1}{n} \left( \pars{1} - \pars{2} + ... + (-1)^{n+1} \pars{n} \right),$$

where $\left\{ x \right\} $ denotes the fractional part of $x$.

My guess is that the limit is equal to 0; I tried finding some asymptotics for the fractional part sum, by looking for example at ways to bound $\pars{k} - \pars{k+1}$; My intuition is that this difference is rather small(perhaps less than $\frac{n}{k(k+1)}$) and that it is big enough to be relevant only when one of them is zero, meaning that $k$ or $k+1$ divides $n$. This would lead me to conjecture that it grows at most as $O(\sqrt{n})$, which would make the limit zero, but I have not been able to make this rigorous. Another idea I had would be to look at the sum with odd denominator and the sum with even denominators and show that they must be "rather" close; this seems pretty intuitive but the fractional part is very chaotic and I have not been able to get any bounds.

Any ideas/tips would be appreciated!