Convergence of an infinite product of a quotient of an irrational number and its decimal expansion

Question

I did some exploring with Mathematica about infinite products of quotients of limits of sums and their quotients with increasing partial sums. I didn't discover much, but I would like to ask the following:

Assume $r \in \mathbb R^+ \setminus \mathbb Q$ (just for simplicity I want to assume that $r$ is positive). Let its decimal expansion be $a_0.a_1 a_2 a_3\dots$.

If we construct an infinite product $\prod_{i=0}^\infty r/a_0.a_1\dots a_i = r/a_0 * r/(a_0.a_1)*r/(a_0.a_1 a_2)\dots$.

Does this product always converge?

Clearly the $i$th quotient is always greater than $1$ and converges to $1$ as $i \rightarrow \infty$. Thus $i$th quotient is of the form $1+q_i$, $q_i>0$. So this product converges if $\sum_{i=0}^\infty q_i$ converges. Does it?

If it converges for some $r$, is the above product always irrational?

Answer 1

Call $t_i = r - a_0.a_1 \ldots a_i$ (the "missing tail"). Then your $q_i$ is just: $$ \begin{align*} q_i &= \frac{r}{r - t_i} - 1 \\ &= \frac{t_i}{r - t_i} \\ &< \frac{10^{-i}}{r - a_0} \end{align*} $$ So the $q$-series converges, and so does your product. To what is anybody's guess, I'm afraid.