power series convergence/divergence

1. Let \sum a_v z^v be an arbitrary power series, and set \varlimsup \sqrt[v]{|a_v|}=\alpha.
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:

    \[r=\fr{\alpha}=\fr{\varlimsup \sqrt[v]{|a_v|}} \qquad \text{(Cauchy-Hadamard formula)}\]


Konrad Knopp – Infinite sequences and series, 1956

Leave a Reply

Your email address will not be published.