If ${\sum }_{n}n!c_n{z}^n$ is entire and $c_n$ does not terminate, then ${\sum }_n{c}_n{z}^n$ has infinitely many zeros

We prove that if ${\sum }_{n}n!c_n{z}^n$ is entire and $c_n$ does not terminate, then ${\sum }_n{c}_n{z}^n$ has infinitely many zeros. We then use this result to give alternative proofs that the Le Roy functions $f_r(z)={\sum }_{n=0}^{\mathrm{\infty }}\frac{z^n}{(n!{)}^r}$ for $r>1$ and Bessel functions $J_{\alpha }(z)={\sum }_{m=0}^{\mathrm{\infty }}\frac{(-1{)}^m}{m!\mathrm{\Gamma }(m+\alpha +1)}{\left(\frac{z}{2}\right)}^{2m+\alpha }$ for $\alpha \in \mathbb{R}$ have infinitely many zeros.