Let $\ x\in\mathbb{R}.\ $ For each $\ n\in\mathbb{N},\ $ define $\ x_n := \displaystyle\min_{k\in\mathbb{Z}}\left\lvert \frac{k}{n} - x \right\rvert.\ $ For which values of $\ x\ $ does $\ \displaystyle\sum_{n=1}^{\infty} x_n\ $ converge?
If $\ x=0,\ $ then $\ x_n = 0\ \forall\ n\in\mathbb{N},\ $ and so in this case, $\ \displaystyle\sum_{n=1}^{\infty} x_n\ = 0 $ and the series converges.
If $\ x\in\mathbb{Q}\setminus\{0\},\ $ then write $\ x = \frac{p}{q}.\ $ Then for any integer $\ m\ $ coprime to $\ q,\ $ let $\ A = \{ km: k\in \left(\mathbb{N}\setminus(q\mathbb{N})\right) \}.\ $ It is not difficult to show that $\ \displaystyle\sum_{n\in A} x_n\ $ diverges by comparison with the harmonic series. Therefore, in this case, $\ \displaystyle\sum_{n=1}^{\infty} x_n\ $ diverges also.
So, I am more interested in if $\ x\not\in\mathbb{Q}.\ $ In this case, does $\ \displaystyle\sum_{n=1}^{\infty} x_n\ $ always converge? Does it always diverge? Or does it depend on the value of $\ x\ ?$
Let $r:=\min_{m \in \mathbb Z} |x-m| >0$. Suppose that $$ x_n=|x-{k/n}|<r/2 \tag{*}$$ for some integers $n,k$ where $n>0$. Then $$\min_{m \in \mathbb Z} |k/n-m| >r/2 \,. $$ For some integer $\ell$, we have $x_{ n+1} =\Big|x-\frac{ k+\ell}{ n+1}\Big|$, so $$x_n+x_{ n+1} \ge \Big|\frac kn-\frac{ k+\ell}{ n+1}\Big|=\frac{|k/n-\ell|}{ n+1 } \ge \frac{r}{2n+2} \,. $$
Thus $$x_n+x_{ n+1} \ge \frac{r}{2n+2} \,. \tag{**}$$ holds for all integers $n>0$, whether or not the assumption $(*)$ holds. Summing $(**)$ over $n>0$ and comparing to a harmonic series, we conclude that $$2\sum_{n>0} x_n \ge \sum_{n>0} \frac{r}{2n+2} =\infty \,.$$