Let $$f(x)= \sum\_{n=1}^{\infty}a\_n x^n$$ where $a\_n\in \{0,1\}$, and the $f(x)$ has a natural boundary. By the way, $$f(x)= \sum\_{n=1}^{\infty}a\_n x^n$$ is rational function or transcendental one on $\mathbb{C}$。


Is $f(x)$ at $\frac{1}{10}$ transcendental? Or is the value of such a transcendental function of power series at $\frac{1}{10}$ a transcendental number?


UPDATE: in addition, let $$f(x)= \sum\_{n=1}^{\infty}a\_n x^n$$ where $a\_n\in \mathbb{N},\exists M\in \mathbb{N}, a\_n < M$, and the $f(x)$ has a natural boundary. Is $f(x)$ at $\frac{1}{10}$ transcendental?



Is the value of the power series at 0.1 transcendental?
This question is likely open.


We can tell whether $f(1/10)$ is rational or irrational (by asking whether $a\_n$ is eventually periodic); in this case, definitely irrational.


Can we have an **algebraic** irrational whose expansion base $10$ consists only of $0$s and $1$s? It is believed that is impossible, but has not been proved. Indeed, it is believed that algebraic irrational numbers are normal in all bases.


Similar question: is there an algebraic irrational point in the middle-thirds Cantor set?


