Infinite linear combination of linearly independent reals equaling 0

Question

Is it possible to find a sequence of real numbers $\{x_n\}$ linearly independent over $\mathbb{Q}$ with the property that there exists a sequence of rationals $\{q_n\}$ such that $\sum q_n x_n = 0,$ but for any sequence of integers $\{a_n\}, \sum a_nx_n \not= 0$?

Answer 1

Of course; take any transcendental number $0<\alpha<1$ and set $x_n:=n(\alpha^{n+1}-\alpha^n)$ and $q_n=\frac{1}{n}$.