For $z\in\mathbb{R}$ it's very easy to show that it holds $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$$ But how do we show the same thing for $z\in\mathbb{C}$
2026-04-15 10:17:20.1776248240
How to show $\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$
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Let $z=re^{i\theta}$. Then $z^n=r^ne^{in\theta}$, and so, $|z^n|=r^n$, and $r \in \mathbb{R}$. So, you can now use your result from the case of reals.