Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

Question

Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

I am able to show this directly by considering $(1-z)P(z)$ and seeing that if $w$ is root of $P(z)$ then $w$ is a root of $(1-z)P(z)$ and then seeing that if $|w|>1$, we get a contradiction.

But I am not sure how to use the Gauss-Lucas theorem to show this. The Gauss-Lucas theorem says: Let $p(z)$ be a polynomial. If $p'(w_0)=0$, then $w_0$ lies in the convex hull of the zeros of $p(z)$.

Answer 1

$P(z) = \frac{d}{dz}(z^n+z^{n-1}+\cdots + z+1)$

$Q(z) = z^n+z^{n-1}+\cdots + z+1 = \frac{z^{n+1}-1}{z-1}$

The zeros of $Q$ lie obviously on the unit disk.

So, the zeros of $P$ lie in the convex hull of the zeros of $Q$, hence also in the unit disc.