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}? $
Complex number inequality?
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inequality
complex-numbers
1 Answers
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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.