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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 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?

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?

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 value of such a transcendental function of power series at $\frac{1}{10}$ a transcendental number?

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 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?

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 $C$。

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

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 value of such a transcendental function of power series at $\frac{1}{10}$ a transcendental number?