Abelian Lemma

The Abelian lemma is an auxiliary result for the investigation of the convergence range of power series . It is named after Niels Henrik Abel .

statement

Be

${\displaystyle P(z)=\sum _{k=0}^{\infty }a_{k}(z-z_{0})^{k}}$

a power series. If there is a point for which the sequence of its summands is limited (in terms of absolute value ), then converges absolutely and normally in the open disc . ${\displaystyle z_{1}\neq z_{0}}$ ${\displaystyle \left\{a_{k}(z_{1}-z_{0})^{k}\right\}_{k}}$ ${\displaystyle P}$ ${\displaystyle \left\{z\in \mathbb {C} \,:\,\left|z_{0}-z\right|<\left|z_{0}-z_{1}\right|\right\}}$

consequence

If one takes into account that the series must always diverge at those points where the sequence of its summands is unbounded (according to the Cauchy criterion for series ), then it follows from the lemma that every power series has a well-defined radius of convergence and on every compact term converges uniformly within the convergence circle, diverges outside the convergence circle. No statement is made about the convergence for points on the convergence circle. ${\displaystyle z\in \mathbb {C} }$

source

• Eberhard Freitag & Rolf Busam: Function theory 1 , Springer-Verlag, Berlin, ISBN 3-540-67641-4 , page 98.