Convergence of an infinite power There are complex numbers $z$ and $w$ for which
$$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$
where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$.
Are there complex numbers $z$ and $w$ for which
$$\lim_{n\rightarrow\infty}z^n=w$$
where $z\neq1$ and $w\neq0$? Prove your conclusion.

No. If $|z|<1$ the limit is $0$, if $|z|>1$ the limit is $\infty$ or does not exist since $|z|^n\to\infty$. If $|z|=1$ then $z=e^{i\theta}$ for $\theta\in[0,2\pi)$ and $z^n=e^{in\theta}$. Unless $\theta=0$, i.e. $z=1$, the sequence will rotate the point on the unit circle by $\theta$ counterclockwise at each step, and there is no limit.
