Consider $f_n(z) = \sum_{k=0}^{n}\frac{z^k}{k!}$. Let $z_n$ be the zero of $f_n$ which is closest to the origin, and prove that $\|z_n\| \geq tn$ for some constant $t$.
First of all, the statement makes perfect sense. Since $f_n$ is the partial sum of the power series expansion of $e^z$ which has no zero at all. So by Hurwitz theorem, the zeros of $f_n$ have to be pushed out. However, I do not know how to make the bound precise.
I do not really think this is the duplicate of the question asked. Since as I explained above using Hurwitz is enough to show that $f_n(z)$ cannot have zeros inside a given compact set for large enough $n$. But here I want a precise bound for $\|z_n\|$.
