Investigating the roots of a polynomial sequence

Question

Define $\large P_n(z) = e^{z} \, \Gamma{\left(n,z\right)}$ where $\large \Gamma \left(n,z\right)$ is the incomplete gamma function.

Alternatively, use the truncated expansions of $\large e^z$ to obtain the same (scaled) sequence of polynomials.

Do the moduli of the roots go to infinity as $\large n$ goes to infinity?

If so, is there a (relatively) elementary proof of this that can be easily understood by a typical high school mathematics student?

Answer 1

Fairly complete information (with references) is given in the answer to this question: Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function