power series convergence/divergence

1. Let $\sum a_v z^v$ be an arbitrary power series, and set $\varlimsup \sqrt[v]{|a_v|}=\alpha$ .
Then:
a) for $\alpha=0$ , the series is everywhere convergent.
b) for $\alpha=+\infty$ , the series is divergent for every $z\neq 0$ .
c) if $0<\alpha<+\infty$ , then the series is absolutely convergent for every $z$ with $|z|<r = \displaystyle\fr{\alpha}$ , divergent for every $z$ with $|z| > r$ .
Thus in all three cases, with suitable interpretation: