given transcendental function
$$F(x)=\sum\_0^{\infty}a\_i x^i,a\_i\in \mathcal{N} \bigcup 0,\exists M \space a\_i \leq M^i$$.


is there algebraic function $$A(x)=\sum\_0^{\infty}b\_i x^i,b\_i\in \mathcal{N} \bigcup 0,$$,such that $a\_i =b\_i$ if $a\_i = 0$;$a\_i \leq b\_i$ otherwise?



is there any algebraic function that has a specific relation to transcendental one?
By Hadamard's theorem, a lacunary series $\sum\_k c\_k z^{\lambda\_k}$ with finite radius of convergence where $\inf\_k \lambda\_{k+1}/\lambda\_k > 1$ can't be analytically continued outside its circle of convergence, and in particular can't be the Maclaurin series of an algebraic function. So if $F(z)$ is such a series, there can be no algebraic $A(z)$. 


