what to do now?
Since power series and stuff belongs at least to calculus II , I assume you've already had calculus I: limits, continuity and stuff...right?
Then you must know that n^(1/n) --> 1 when n --> oo, but then
n^(2/n) = [n^(1/n)]^2 --> 1*1 = 1 by arithmetic of limits.
Hmmm...rechecking your question I realize that what you posted is NOT a power series in z since z appears in the denominator, so this is at most a Laurent series, a very different stuff belonging to complex analysis.
What did you really mean to ask here? Is z a variable or just a parameter?
True, because as I pointed out before what you ngave is not a power series! What you gave is a palin series with a parameter (constant) z, as you said. There's no convergence radius here, then.
For something to be a proper power series the powers of the VARIABLE have to be positive integers.