  We extend the notion of rational ergodicity to $\beta$-rational ergodicity
for $\beta > 1$. Given $\beta \in \mathbb R$ such that $\beta > 1$, we
construct an uncountable family of rank-one infinite measure preserving
transformations that are weakly rationally ergodic, but are not
$\beta$-rationally ergodic. The established notion of rational ergodicity
corresponds to 2-rational ergodicity. Thus, this paper answers an open question
by showing that weak rational ergodicity does not imply rational ergodicity.
