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Find all z such that the series nnzn converges

Find all complex numbers z such that the series

    \[ \sum_{n=1}^{\infty} n^n z^n \]

converges.


We consider the limit of the nth root of the terms of the series,

    \begin{align*}  \lim_{n \to \infty} (a_n)^{\frac{1}{n}} &= \lim_{n \to \infty} \left( n^n z^n \right)^{\frac{1}{n}} \\[9pt]  &= \lim_{n \to \infty} nz \\[9pt]  &= \begin{dcases} +\infty &\text{if } z \neq 0 \\ 0 & \text{if } z = 0. \end{dcases} \end{align*}

Hence, the series converges if and only if z = 0.

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