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Suppose $|z|>1$ for $z$ a complex number. I'm trying to build a certain comparison test to test convergence. I'm wondering, is it true that $$ \frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1}? $$

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up vote 8 down vote accepted

Yes. By triangle inequality, we have $$|z|^n=|z^n|=|z^n+1-1|\leq|z^n+1|+1.$$ This implies that $$|z|^n-1\leq |z^n+1|.$$ Since $|z|>1$, we have $0<|z|^n-1\leq |z^n+1|$. Therefore, $$\frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1},$$ as required.

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