  We generalize the classical Olivier's theorem which says that for any
convergent series $\sum_n a_n$ with positive nonincreasing real terms the
sequence $(n a_n)$ tends to zero. Our results encompass many known
generalizations of Olivier's theorem and give some new instances. The
generalizations are done in two directions: we either drop the monotonicity
assumption completely or we relax it to the monotonicity on a large set of
indices. In both cases, the convergence of $(na_n)$ is replaced by ideal
convergence.
  In the second part of the paper, we examine families of sequences for which
the assertions of our generalizations of Olivier's theorem fail. Here, we are
interested in finding large linear and algebraic substructures in these
families.
