Find all complex numbers 
such that the series
![\[ \sum_{n=1}^{\infty} \frac{z^n}{3^n} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-0875d4a3a865e2754e2403c26441c423_l3.png "Rendered by QuickLaTeX.com")
converges.
We can write
![\[ \sum_{n=1}^{\infty} \left|\frac{z^n}{3^n}\right| = \sum_{n=1}^{\infty} \left|\left( \frac{z}{3} \right)^n\right|. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-11575a0968b97fc22a77d6ffc6c2b9e7_l3.png "Rendered by QuickLaTeX.com")
This is a geometric series, and so converges absolutely for 
which implies 
, and diverges for 
.
Find all complex numbers such that the series
converges.
We can write
This is a geometric series, and so converges absolutely for which implies , and diverges for .