# Convergence of complex series that are not absolutely convergent?

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

• This seems very vague to me. Why don't you describe the particular series? – Yemon Choi Sep 19 '14 at 21:55
• I need something stronger than the Dirichlet test, since my series fails it, but still converges! Are such examples known? – André LeClair Sep 20 '14 at 17:35
• Somehow, this got put on hold, but whoever suggested summation by parts, thank you! It did the trick. – André LeClair Oct 15 '14 at 18:14

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 $x\neq0$).