I found the following definition of a Cantor Product expansion of $x \in [0, 1)$ in a book:
Let $r(x) = \lfloor \frac{x}{1 - x} \rfloor$, and define $T(x) = x \frac{r(x) + 1}{r(x)}$. Then (I'm told)
\begin{align*} x & = \prod_{k = 0}^{\infty} \frac{r_{k}(x)}{r_{k}(x) + 1} \end{align*}
where $r_{k} = r \circ T^{k}$.
Here's what I have.
\begin{align*} x & = T^{K + 1}(x) \prod_{k = 0}^{K} \frac{r_{k}(x)}{r_{k}(x) + 1} \end{align*}
I think it should work if $T^{K}(x) \to 1$, but am not sure. Could somebody familiar with this please help out? Thanks.