We can identify absolutely convergent series with the $l^1$ space and conditionally convergent series with a subspace of $c_0 = \{\{a_n\} \in l^\infty : a_n \to 0\}$ Since $l^1$ contains all finite sequences, it is dense in $c_0$, so in this sense absolutely convergent series are dense, or "common" among all series. However, this result is not terribly illuminating, since finite sequences are also dense in $c_0$.

Are there other ways to look at the relationship between absolutely and conditionally convergent series? Maybe absolutely convergent series are meager, open or dense when we consider a more appropriate topology on the set of convergent sequences.

  • 2
    $\begingroup$ It's not an answer, but it's worth pointing out that in the case of power series, the interior of the disk of convergence guarantees absolute convergence, whereas conditional convergence can only occur on the boundary sphere, if at all. In that sense, conditional convergence appears to be the rarer condition. $\endgroup$ – Theo Bendit Nov 5 '17 at 4:39
  • $\begingroup$ Like you wrote in the answer, if a sequence is the limit of its finite parts then $l^1$ is dense. I can't think of any natural topology that doesn't satisfy that. $\endgroup$ – Petr Naryshkin Nov 5 '17 at 22:11
  • $\begingroup$ Proper subspaces of Banach spaces are always of first category. $\endgroup$ – Jochen Nov 6 '17 at 12:26

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