In this post we denote the primorial of order $n$ as $N_n=\prod_{k=1}^n p_k,$ where thus $p_k$ denotes the $k$th prime number.
This section of the Wikipedia's article dedicated to transcendental numbers tell us some classic examples of transcendental numbers defined as infinite series and involving certain powers of two or well factorial numbers. I wondered if is in the literarature some series (I evoke similar than previous ones, see the example below) involving primorials.
For example I wondered if is feasible to prove or refute that $$\sum_{n=1}^\infty 10^{-N_n}\tag{1}$$ is a transcendental number.
Question. I would like to know* if is in the literature that $\sum_{n=1}^\infty 10^{-N_n}$ is a transcendental number. In this case please refer the literature and I try to find and read such fact. In case that it isn't known please add what work can be done about the deduction concerning to if $\sum_{n=1}^\infty 10^{-N_n}$ can be proven/disproven as a trasncendental number. Many thanks.
*I've no intutition about if $(1)$ can be a transcendental number.
Let $x = \sum_{n=1}^\infty 10^{-N_n}$ and $x_m = \sum_{n=1}^m 10^{-N_n}$. Then $x_m$ is rational with denominator $10^{N_m}$. Since for any positive integer $k$, $$|x - x_m| \le 2 \cdot 10^{-N_{m+1}} < \left(10^{N_m}\right)^{-k}$$ for sufficiently large $m$, $x$ is a Liouville number and therefore is transcendental by Liouville's theorem.