Sei ${\displaystyle {}z\in \mathbb {C} }$, ${\displaystyle {}\vert {z}\vert <1}$. Berechne zur summierbaren Familie

${\displaystyle z^{k}z^{\ell },\,(k,\ell )\in \mathbb {N} ^{2},}$

die Teilsummen

${\displaystyle s_{k}=\sum _{\ell \in \mathbb {N} }z^{k}z^{\ell }}$

${\displaystyle {}k\in \mathbb {N} }$

${\displaystyle {}\sum _{k\in \mathbb {N} }s_{k}}$