If $q_n$ be an enumeration of rational numbers, for which $x$ the following sequence converges?
$$\sum_{n=1}^{\infty}e^{-n^2|x-q_n|}.$$
I guess that for no $x$ the sequence converges. I tried to consider a subsequence of $q_n$ converging to $x$ such that $e^{-n_k^2|x-q_{n_k}|}$ does not converge to zero.