Exercise 2, § 8

Show that, when $z$ varies in the unit disc $|z| \leq 1$ in $\mathbf{C}$, then the convex cone generated by the points $(z, z^2, ..., z^n)$ in the spaces $\mathbf{C}^n$, is the whole of the space $\mathbf{C}^n$. (Note that there cannot exist complex numbers $c_k$ not all zero $(1 \leq k \leq n)$ such that $\mathscr{R} \left( \sum_{k=1}^n c_k e^{ki\theta} \right) \geq 0$ for $0 \leq \theta \leq 2\pi$, using the fact that $\int_0^{2\pi} e^{ki\theta} d\theta = 0$ for every integer $k \neq 0$.)