  Given a non-negative real sequence $\{c_n\}_n$ such that the series
$\sum_{n=1}^{\infty}c_n$ diverges, it is known that the size of an infinite
subset $A\subset\mathbb{N}$ can be measured in terms of the linear density such
that the sub-series $\sum_{n\in A}c_n$ either (a) converges or (b) still
diverges. The purpose of this research is to study these convergence/divergence
questions by measuring the size of the set $A\subset\mathbb{N}$ in a more
precise way in terms of the recently introduced asymptotic $\psi$-density. The
convergence of the associated sub-signed series $\sum_{n=1 }^{\infty}m_nc_n$ is
also discussed, where $\{m_n\}_n$ is a real sequence with values restricted to
the set $\{-1, 0, 1\}$.
