I was looking for a product analog of Engel expansion https://en.wikipedia.org/wiki/Engel_expansion and came up with this new type of expansion of a number.
We start with a number $0<x_1<1$.
Then find integer $n_1$ such that $$ 1-\frac{1}{n_1-1}<x_1<1-\frac{1}{n_1}, $$ Then write $$ x_1=\left(1-\frac{1}{n_1}\right)x_2, \quad 0<x_2<1. $$ Then continue the process using recurrence $$ n_k=1+\lfloor\frac{1}{1-x_{k}}\rfloor. $$ For example we get $$ \frac{\pi}{4}=\left(1-\frac{1}{5}\right) \left(1-\frac{1}{55}\right) \left(1-\frac{1}{13931}\right) \left(1-\frac{1}{2811273900}\right)\\\times \left(1-\frac{1}{75684454671917856333}\right) \left(1-\frac{1}{17449196642525545927189430934592764076742}\right)\times \ldots $$
I looked this sequence in oeis but it did not return any results.
Also it seems like $(n_k)^{2^{-k}}$ should have a limit, but the numebrs grow very fast and this limit does not converge fast enough to calculate or check if it really tends to some limit.
Q: has representations of this kind been studied before? Can you prove the limit exits and find the limit $\lim(n_k)^{2^{-k}}$, $k\to\infty$?