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

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

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?