I'm working on a Homework problem involving Convergence of Random variables and I've arrived at an expression which looks like follows:
$$ M_{X_n}(ju)= \prod_{k=1}^{n}0.5\frac{1-e^{\frac{ju}{2^{k-1}}}}{1-e^\frac{ju}{2^k}}$$
,where $M_X(ju)$ denotes the characteristic function of a random variable $X$. It'd be great to have an hint to proceed as I've not seen(or remember) a product like this before and I'd love to show my work here but this is just a single step of a big homework problem and I'm just stuck at this part. Unfortunately, I don't seem to have too many thoughts about this particular step. (I know some work has to be shown, but really I don't seem to find a way through this step and that's the reason why I'm posting this; Thank You !!)
Edit: I made an error in the initial equation and I've rectified it. $0.5$ should have been inside the product.
Hint: if $e^{ju/2^{k}} = t$, then $t^2 = e^{ju/2^{k-1}}$