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Find all z such that the series (z / (2z+1))n converges

Find all complex numbers z such that the series

    \[ \sum_{n=1}^{\infty} \left( \frac{z}{2z+1} \right)^n \]

converges.


This is a geometric series. It converges if

    \[ \left| \frac{z}{2z+1} \right| < 1 \quad \implies \quad \left| 2 + \frac{1}{|z|} \right| > 1 \]

and diverges if

    \[ \left| 2 + \frac{1}{|z|}\right| < 1. \]

If \left| 2 + \frac{1}{|z|} \right| = 1 then the series also diverges since \lim_{n \to \infty} a_n \neq 0 in this case.

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