Any closed form for series like $$F(x)=\sum_{p=2}^{\infty}x^p,\quad p\text{ is prime}$$ or $$F(x)=\sum_{i=0}^{\infty}x^{i!}\quad ?$$

More generally, we can obtain a power series from decimal expansion of a number $r$ (with $0< r<1$) by replacing $$\left(\frac{1}{10}\right)^i$$ with $$x^i$$ like $$\frac{1}{3}=3\left(\frac{1}{10}\right)^1+3\left(\frac{1}{10}\right)^2+\dotsb 3\left(\frac{1}{10}\right)^i+\dotsb.$$ In this example we obtain: $$f(x)=\sum_{i=1}^{\infty}3x^i.$$

- When $f(x)$ is convergent, what restriction do we have to put on $r$ (if $r$ is c.e. number) to make $f(x)$ have a closed form?
- When is $f(x)$ algebraic, or transcendental?

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