Given a series with integral coefficiens as following:
$$F(x)=\sum\_0^i a\_i x^i,\text{where }a\_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i \psi(i) =a\_i$$
Is there any algorithm to decide whether such a series is a algebraic function?



Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function?
If you are given a Turing machine and an input to it, define $a\_i$ as follows. First $a\_i=0$ if $i$ is not a power of $3$. If the Turing machine hasn't halted after $n$ steps set $a\_{3^n} = 1$ otherwise if it halts set $a\_{3^j}=0, j \ge n$. These coefficients are the values of a computable function. As the series $\sum\_0^{\infty} x^{3^n}$ is transcendental, having an algorithm that for all such data determines whether the resulting series is algebraic or not is equivalent to solving the Halting problem, which is known to be unsolvable. 


