Suppose that $\alpha_1, ..., \alpha_k, z_1, ..., z_k\in\mathbb{C}$ and $|z_1| = |z_2| = \cdots = |z_k|$.
For $k > 1$, what can be said about the limit
$$\lim_{n\rightarrow\infty} \frac{\alpha_1z_1^{n+1} + ... + \alpha_kz_k^{n+1}}{\alpha_1 z_1^n + ... + \alpha_kz_k^n}$$