Is $e$ a zero of power series with rational coefficients?

Since $e$ is a transcendence number, so it is certain that it is not zero of any polynomial with rational coefficients. However, I wonder can we find a power series with rational coefficient such that it is zero evaluated at $e$. If such series exists, can we explicitly find the coefficients?

Yes. You can build such a power series recursively.

Start with $a_0 = 1$ (or anywhere else you like).

Now find a rational multiple $a_1$ of $e$ such that $$ 0 < 1 + a_1e < 1/2. $$ Then find rational $a_2$ such that $$ 0 < 1 + a_1e + a_2e^2< 1/4. $$ Continue in the obvious way.

Clearly this procedure produces many such series. I don't know whether there's one that's particularly nice.