$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=+\infty$$ Can one conclude that $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an irrational number? a transcendental number?
A special case is $a_n=n!$, $e$ is a transcendental number.
Another special example is Liouville number $\sum\limits_{n=1}^\infty\dfrac1{10^{n!}}$ is a transcendental number, too.
so the question, if true, may be difficult.
The question is a generalization of If $(a_n)$ is increasing and $\lim_{n\to\infty}\frac{a_{n+1}}{a_1\dotsb a_n}=+\infty$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is irrational