Does anyone know of a convergence test for a complex series of the form

$$\sum_n a_n \cdot \exp(i \cdot b_n)$$

?

The particular series I need to understand has a_n going to zero as n goes to infinity, but it fails the absolute convergence test. However numerically I do find it to converge. It should have something to do with convergence of Fourier series (since $\ \exp(i\cdot t)\ =\ \cos(t)\,+\,i\cdot \sin(t)$).