A sum with strong oscillations

I encounter a sum in the form

\begin{equation}\sum_{j=1}^{\infty}a_{j}b_{j}e^{i j\theta}\end{equation}

I expect a lot of cancellations to happen due to the oscillatory term $e^{ij\theta}$.I can control the sum\begin{equation}\sum _{j=1}^{\infty}a_{j}e^{i j\theta}\end{equation}by the method of partial summations as I can use cancellations due to having control on the sequence $a_{j+1}-a_{j}$.

However I know $b_{j}$ s are bounded and positive.

Can someone tell me how to tackle such an issue?