how to determine the radius of convergence of the power series
Σ 2^(n) z^(nē).
thanks in advance .regards.
Treat it as an ordinary series (rather than a power series) and use the ratio test:
$\displaystyle \frac{(n+1)\text{th term}}{n\text{th term}} = \frac{2^{n+1}z^{(n+1)^2}}{2^nz^{n^2}} = 2z^{2n+1}.$
The limit of that as $\displaystyle n\to\infty$ will depend on whether or not $\displaystyle |z|<1.$