I've been curious about methods of proving the irrationality of some infinite products and had this idea.
Suppose
$$\prod_{n=1}^ka_n=\frac{b_k}{c_k}$$
where $a_n\in\mathbb{Q}$ and $\gcd(b_k,c_k)=1$. If the number of factors in $b_k$ and $c_k$ are absolutely increasing as $k$ increases, is it possible for the product to converge to a rational? My intuition says not, however the nature of infinity can be quite tricky.
In other words, if $\sigma$ is the divisor function, and for all $k$, we have that $\sigma_0(b_{k+1})>\sigma_0(b_k)$ and $\sigma_0(c_{k+1})>\sigma_0(c_k)$, must the infinite product be irrational? If so, if the restrictions were less tight, such that forall $k$ there exists $m>k$ such that $\sigma_0(b_m)>\sigma_0(b_k)$ and $\sigma_0(c_m)>\sigma_0(c_k)$, does that change anything?
Let me know your thoughts.
If you take $a_n =\frac{2}{2n+1} $ then obviously the product converges to $0$