  A sequence $\{g_k\}_{k=1}^\infty$ in a Hilbert space $\cal H$ has the
expansion property if each $f\in \overline{\text{span}} \{g_k\}_{k=1}^\infty$
has a representation $f= \sum_{k=1}^\infty c_k g_k$ for some scalar
coefficients $c_k.$ In this paper we analyze the question whether there exist
small norm-perturbations of $\{g_k\}_{k=1}^\infty$ which allow to represent all
$f\in \cal H;$ the answer turns out to be yes for frame sequences and Riesz
sequences, but no for general basic sequences. The insight gained from the
analysis is used to address a somewhat dual question, namely, whether it is
possible to remove redundancy from a sequence with the expansion property via
small norm-perturbations; we prove that the answer is yes for frames
$\{g_k\}_{k=1}^\infty$ such that $g_k\to 0$ as $k\to \infty,$ as well as for
frames with finite excess. This particular question is motivated by recent
progress in dynamical sampling.
