I'm trying to find the radius of convergence and the area of convergence of the following power series :
$$\sum_{n=0}^{\infty } (n+1/2)z^n$$
The radius of convergence is easily found to be 1 by the ratio test. So inside the unit circle we have convergence.
But how about the entire area of convergence? It's not like in the real case where you can just plug in the interval end points and thereby know if these are in the area of convergence or not. There seems to be a infinite amounts of points on the unit circle to deal with in this complex case? Is there some trick involved here?
Best Answer
The series cannot possibly converge when $|z|\geq1$, since its terms don't go to zero.