# For which x the following sequences converges

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.

• Integrating over $\mathbb R$ we see that the series converges for almost all $x$. – Kavi Rama Murthy Jan 7 '19 at 23:53
• What kind of answer are you looking for? "For what $x$" is very unclear. Do you have to show that there can only be countably many $x$ where it diverges, for example? – A. Pongrácz Jan 8 '19 at 0:42