Is there an entire function with $f(\mathbb{Q}) \subset \mathbb{Q}$ and a non-finite power series representation having only rational Coeffitients I'm trying to answer the following question:
Is there an entire function $f(z) := \sum \limits_{n=0}^\infty c_nz^n$ such that


*

*$f(\mathbb{Q}) \subset \mathbb{Q}$

*$\forall n: c_n \in \mathbb{Q}$

*$f$ is not a polynomial
?
I'm trying to show that no such function exists. Here's why I think so:
Assuming such a function existed. We would get $f(10^k) \in \mathbb{Q}$ for all $k \in \mathbb{Z}$. So the decimal representation of $f(10^k)$ either cuts at some digit or consists of repeating digits.
Now my gut is telling me that if this is true for $f(10^n)$ with $n \in \mathbb{N}$, it won't be for $f(10^{-n}).$ (e.g. for $c_n$ with a finite digit representation: that's because the number of zeroes between each non-zero digit would increase indefinitely)
But, is this correct at all? And if so, how do I show it rigorously?

If you allow meromorphic functions (and as a consequence, finite radius of convergence), you have
$$
\frac{1}{1-x}=1+x+x^2+x^3+\cdots
$$
