Can a power series with rational cffs. that sum to irrational lim evaluate to rational lim at non-zero rational point?

Question

Assume we have $f(x) = \sum a_n x^n, a_n \in \mathbb{Q}$, and convergent $f(1) \notin \mathbb{Q}$. Assuming $f(x)$ converges at some $f(x \in \mathbb{Q})$, is it possible for $f(x \in \mathbb{Q}) \in \mathbb{Q}$ for some non-zero $x$?

Thanks!

Answer 1

Consider $$ \sqrt{1+\frac{x}{2}}=\sum_{k\ge0}\binom{1/2}{k}\frac{1}{2^k}x^k $$ that converges to $\sqrt{3/2}$ for $x=1$.

Take $x=-3/2$; then the series converges to $\sqrt{1/4}=1/2$.