Suppose the series $ \sum_{n=1}^{\infty} a_n$ converges in $ \mathbb{Q}$ i.e., $ \sum_{n=1}^{\infty} a_n \in \mathbb{Q}$ , where $ a_n$ is arbitrary rational numbers. Consider the power series $ \sum_{n=1}^{\infty} a_nx^n$ . What condition on the variable $ x$ ensures that the power series $ \sum_{n=1}^{\infty} a_nx^n \in \mathbb{Q}$ ?

**Answer:**

For particular case,

$ a_n=\frac{1}{p^n}$ , for some prime $ p$ .

Then the series $ \sum_{n=1}^{\infty} a_n=\sum_{n=1}^{\infty} \frac{1}{p^n}$ converges in $ \mathbb{Q}$ and in fact $ \sum_{n=1}^{\infty} \frac{1}{p^n}=\frac{1}{p-1} \in \mathbb{Q}$ .

Next consider the corresponding power series $ \sum_{n=1}^{\infty} \frac{x^n}{p^n}$ , it has sum $ S(x)=\frac{x}{p-x}$ .

Now $ S(x) \in \mathbb{Q} \Rightarrow S(x)=r \in \mathbb{Q} \Rightarrow x=S^{-1}(r), \ r \in \mathbb{Q} $ .

This is particular case.

Does there exists any general theory solving my above question where $ a_n$ is arbitrary rational number but gives condition on the variable or argument $ x$ ?

This concept is needed in my research.

Please someone help