Find all complex numbers 
such that the series
![\[ \sum_{n=1}^{\infty} \frac{z^n}{n^n} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-3276af28c39016e15521af96f433c2ad_l3.png "Rendered by QuickLaTeX.com")
converges.
First, we have
![\[ \sum_{n=1}^{\infty} \frac{z^n}{n^n} = \sum_{n=1}^{\infty} \left( \frac{z}{n} \right)^n. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-2af57e1d6963407ea6a28b731256a59c_l3.png "Rendered by QuickLaTeX.com")
Then, we look at the limit as 
of the 
th root of the terms,
![\begin{align*} \lim_{n \to \infty} \left( \frac{z}{n} \right)^n &= \lim_{n \to \infty} \frac{z}{n} \\[9pt] &= 0 \qquad \text{for all } z \in \mathbb{C}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1facdba21764d3b2ff97a1805c455dca_l3.png "Rendered by QuickLaTeX.com")
Hence, the series is absolutely convergent for all 
.
Find all complex numbers such that the series
converges.
First, we have
Then, we look at the limit as of the th root of the terms,
Hence, the series is absolutely convergent for all .