The centroid of the zeros of the kth partial sum of exp(z) is -1?

Question

The question is the next:
Let $P_k=1+z+\frac{z^{2}}{2!}+...+\frac{z^{k}}{k!}$, the kth partial sum of $e^{z}$.
(a) Show that, for all values of $k\geq 1$, the centroid of the zeros of $P_{k}$ is -1.
(b) Let $z_{k}$ be a zero of $P_{k}$ with maximal possible absolute value. Prove that $\left \{ \left | z_{k} \right | \right \}$ is an increasing sequence.
I´m worried mainly about the part (a), because I´ve tried to prove it by induction, but I´m not sure how to show the last step of the induction.
PS: Let $\frac{z_{1}+z_{2}+...+z_{n}}{n}$ denote the centroid of the complex numbers $z_{1},z_{2},...z_{n}$.
PS2: I think that this fact can help: "The centroid of the zeros of a polynomial is the same as the centroid of the zeros of its derivative".

Answer 1

A) About your first question: you can attempt an induction proof but it is simpler to make a direct use of one of the Vieta's formulas: the sum of the roots $r_1,\cdots r_n$ of $\sum_{k=0}^n a_kz^k=0$ is equal to

$$\sum_{k=1}^n r_k=-\frac{a_{n-1}}{a_n}\tag{1}$$

Knowing that here $a_k=\dfrac{1}{k!}$, (1) becomes, divided by $n$:

$$\dfrac{1}{n}\left(\sum_{k=1}^n r_k\right)=-\dfrac{1}{n}\dfrac{1}{(n-1)!}n! =-1$$

B) The essential property

$$P'_k(z)=P_{k-1}(z)$$

has an immediate consequence: the "russian dolls" inclusion of the successive convex hulls due to the classical Gauss-Lucas theorem. Here is a graphical representation of the roots and their convex hulls; the result is rather amazing (see below the Matlab program that has generated this figure)

Successive convex hulls of the roots of the $P_k$s, from $k=2$ to $k=25$ (the largest one). "Central" point $(-1,0)$ is materialized by a star.

This figure is of course symmetrical with respect to $x$ axis, due to the fact that roots of polynomials with real coefficients are either real or come in conjugate pairs.

Other properties can as well be observed, deserving further analysis, dealing in particular with your second question about the modulus of the roots with largest modulus (situated on a curve very close to a parabola).

Much more can be said about these roots. A fundamental article here recapitulates numerous results, in particular connected to Lambert's function $W$; observe the final semigraphical representation, surely the best one could obtain with printers 50 years ago, a testimony of the state of the art at that time, even with IBM labs top equipment...

See as well this StackExchange question and its answers.

Matlab program:

clear all;close all,hold on;
plot(-1,0,'pk');
plot([-1,-1],[-1,1],'b');C=[1/2,1,1];% case k=2
n=25;
for k=3:n; % k=degree
    C=[C(1)/k,C];
    r=roots(C);R=real(r);I=imag(r);
    H=convhull(R,I);
    plot(R(H),I(H),'b');
    plot(R,I,'.r');
end