We can create a model of an infinite one-dimensional ionic crystal. Considering a system of $N\gg1$ alternating point charges $Q$ and $-Q$, that are distributed as the distance between two neighboring charges is given by $a$.
If we calculate the potential on a spot of a positive charge will get
$$V=2\left(-\dfrac{kQ}{a}+\dfrac{kQ}{2a}-\dfrac{kQ}{3a}+\dfrac{kQ}{4a}-...\right)$$
$$V=-\dfrac{2kQ}{a}\left(1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}-...\right)$$
Then our challange is calculate the sum
$$S=\sum^\infty_1\dfrac{(-1)^{n+1}}{n}$$
If we make the sum in the following order
$$S=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}-...$$
We get
$$S=\ln2$$
However the sum $S$ is a conditionally convergent serie and by Riemann's theorem the result of the sum depend on the order that we make, for example, if we get
$$S=1+\dfrac{1}{3}+\dfrac{1}{5}+\dfrac{1}{7}-\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{11}+\dfrac{1}{13}+\dfrac{1}{15}-\dfrac{1}{6}-\dfrac{1}{8}+...$$
The result is
$$S=\dfrac{3}{2}\ln2$$
For more details of Riemann's theorem and alternating divergent series, see this link.
Then the potential in a spot depends on how i makes the sum? How to explain physically this result and which value i would find if i make the experiment?
For a line of charge, the potential difference between points at finite and infinite radius is $\infty$. See AccidentalFourierTransform's answer. The divergence is the same for a line of point charges, so the difference is indeterminate.
Infinite systems are unphysical idealizations. However, if you start with finite $N$ and keep adding charges equally to both ends, you should get a finite limit.