If your $a_n$s are nonnegative, decrease monotonically, and approach zero, and the $b_n$s are such that $\sum e^{i x b_n}$ remains bounded, then Dirchlet's test would apply here and give you convergence. If the $b_n$'s are an arithmetic sequence, as in the case of Fourier series, then you have the second of these criteria (for $x\neq0$).

If your $a_n$s are nonnegative, decrease monotonically, and approach zero, and the $b_n$s are such that $\sum e^{i x b_n}$ remains bounded, then Dirichlet's test would apply here and give you convergence. If the $b_n$'s are an arithmetic sequence, as in the case of Fourier series, then you have the second of these criteria (for $x\neq0$).