  This article is the natural continuation of the paper: Mukhammadiev A.~et al
Supremum, infimum and hyperlimits of Colombeau generalized numbers in this
journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a
classical series $\sum_{n=0}^{+\infty}a_{n}$ of generalized numbers
$a_{n}\in\tilde{R}$ is convergent if and only if $a_{n}\to0$ in the sharp
topology. Therefore, this property does not permit us to generalize several
classical results, mainly in the study of analytic generalized functions (as
well as, e.g., in the study of sigma-additivity in integration of generalized
functions). Introducing the notion of hyperseries, we solve this problem
recovering classical examples of analytic functions as well as several
classical results.
