TITLE: Lower Bounds for the Roots of Polynomials
QUESTION [5 upvotes]: I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function. 
For example, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are 
$$\mathrm P_k(z) = 1 + z + \frac{1}{2!}z^2 + \cdots + \frac{1}{k!}z^k$$
I want to find the best possible lower bound for the modulus of the roots for any given $\mathrm P_k$. Let
$$ L_k := \min\{ |w| : \mathrm P_k(w)=0 \}$$
I would like to know the limit of $L_k$ as $k \to \infty$. In this case, I think that $L_k \to \infty$ and $k \to \infty$.
I've used the standard bound based on the Rouché theorem, but this does not give a useful lower bound; it gives a lower bound independent of $k$. In the case of $\mathrm e^z$, the lower bound is $\frac{1}{2}$. 
I was hoping for a bound dependent on $k$, so that I could then find the limit of the bound as $k \to \infty$ and show that $L_k \to \infty$.
In general: Given a sequence of polynomials, how could I find the greatest lower bound of the modulus of the roots for each of these polynomials? Specifically, I am interested in the limit of these lower bounds as the polynomials tend to an infinite power series.

REPLY [2 votes]: Your question is related to the Szego curve, which has been much studied (just google it). A lower bound for the roots is given by $W(1/e)n=0.278..n$ where $W$ is the Lambert function.
Theorem 4 of http://www.math.kent.edu/~varga/pub/paper_184.pdf should give you an expansion of a lower bound, up to order $O(1/n)$.