Let \(S^\ast\) be the set of all entire \(f(z)=\sum_{k=0}^\infty a_kz^k\) such that \(a_k>0\) and that an unbounded sequence \(\{n_k\}\) exists such that the \(n_k\) partial sums \(S_{n_k}\) have only real zeros. For a given \(f\) as above, define the sequence \[ q_n(f) := \frac{a_{n-1}^2}{a_{n-2}a_n} \] for \(n\geq 2\). The main result of the article states that if \(f\in S^\ast\) and if the limit \(q_0:=\lim_{n\to\infty} q_n(f)\) exists, then for every positive integer \(m\) we have that the polynomial \[ \sum_{k=0}^m \frac{z^k}{k!(\sqrt{q_0})^{k^2}} \] only has real roots. As a corollary of this and other observations, the authors prove that if \(f\in S^\ast\) and if the limit \(q_0:=\lim_{n\to\infty} q_n(f)\) exists, then \(q_0\geq q_\infty\), where \(q_\infty \approx 3.23\).