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 $\ \exp(i\cdot t)\ =\ \cos(t)\,+\,i\cdot \sin(t)$). 



Convergence of complex series that are not absolutely convergent?
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 $x\neq0$).


