Find all complex numbers 
such that the series
![\[ \sum_{n=0}^{\infty} \frac{(z-1)^n}{(n+2)!} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7f886a77d70c14da7c0c0ab3ade5c6a9_l3.png "Rendered by QuickLaTeX.com")
converges.
Consider,
![\begin{align*} \lim_{n \to \infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to \infty} \left( \frac{(z-1)^{n+1}}{(n+3)!} \right) \left( \frac{(n+2)!}{(z-1)^n} \right) \\[9pt] &= \lim_{n \to \infty} \frac{z-1}{n+3} \\[9pt] &= 0 \qquad \text{for all }z \in \mathbb{C}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1404503ef55d85163f457a63e90a3102_l3.png "Rendered by QuickLaTeX.com")
Hence, the series converges for all complex .
Find all complex numbers such that the series
converges.
Consider,
Hence, the series converges for all complex .