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

Find all complex numbers z such that the series

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

converges.


This is a geometric series

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

Therefore it converge if and only if

    \[ \frac{1}{1+|z|^2} < 1 \quad \implies \quad |z|^2 > 0 \quad \implies \quad z \neq 0. \]

Hence, the series converges for all z \in \mathbb{C} except z = 0.

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