I was posed this problem, it took me a while to solve it – but, I did nevertheless. I shall pose it for all of you, too. In my opinion it is a great exercise.
For what values of $x$ is the series convergent: $$\sum_{n = 1}^\infty (nx) \prod _{k=1}^n \left(\frac{\sin(ka)^2}{1 + x^2 + \cos(ka)^2}\right)$$
Okay, no one has answered -- so, I feel obligated to post my solution to those who read this and do not know how to solve.
We notice initially that for x = 0 we have everything equal to zero, which is convergent and a part of the domain of solutions. If we have $\sin(ka) = 0$, then we have a product of just zeroes, which leads to a convergent value. So, for x and $\sin(ka)$ not equal to 0 we just have the infinite series. From here we can use the ratio test, then the direct comparison test to state that all values of x lead to a convergent series. In conclusion, the domain of solutions is all real numbers.