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](https://en.wikipedia.org/wiki/Properties_of_polynomial_roots#Based_on_the_Rouch.C3.A9_theorem) 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.



Lower Bounds for the Roots of Polynomials
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 $O(1/n)$.


