Determine the radius of convergence 
of the power series:
![\[ \sum_{n=1}^{\infty} (1 - (-2)^n)z^n. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b73d352835d232932ebe6aff27ccd217_l3.png "Rendered by QuickLaTeX.com")
Test for convergence at the boundary points if is finite.
We have
![\begin{align*} \sum_{n=1}^{\infty} \left| (1 - (-2)^n)z^n \right| &= \sum_{n=1}^{\infty} \left|2^{n+1} \right| \left|z^n\right| \\[9pt] &= 2 \sum_{n=1}^{\infty} \left| 2z \right|^n. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-328109223af86ffabc1cf1449bee8d99_l3.png "Rendered by QuickLaTeX.com")
This is a geometric series which converges if and only if 
which implies 
. Hence, 
, and this converges at none of the boundary points.
carteou full
for n=1, the equation doesn’t match, even though this mistake leads to the correct answer.