Editing Series (mathematics) (section)

=== Unconditionally convergent series ===
Suppose that <math>I = \N.</math>  If a family <math>a_n, n \in \N,</math> is unconditionally summable in a Hausdorff [[abelian topological group]] <math>X,</math> then the series in the usual sense converges and has the same sum,

<math display=block>\sum_{n=0}^\infty a_n = \sum_{n \in \N} a_n.</math>

By nature, the definition of unconditional summability is insensitive to the order of the summation.  When <math>\textstyle \sum a_n</math> is unconditionally summable, then the series remains convergent after any [[permutation]] <math>\sigma : \N \to \N</math> of the set <math>\N</math> of indices, with the same sum,

<math display=block>\sum_{n=0}^\infty a_{\sigma(n)} = \sum_{n=0}^\infty a_n.</math>

Conversely, if every permutation of a series <math>\textstyle \sum a_n</math> converges, then the series is unconditionally convergent.  When <math>X</math> is [[Complete topological group|complete]] then unconditional convergence is also equivalent to the fact that all subseries are convergent; if <math>X</math> is a [[Banach space]], this is equivalent to say that for every sequence of signs <math>\varepsilon_n = \pm 1</math><!-- this is not about convergence of functions, even less about uniform convergence. -->, the series

<math display=block>\sum_{n=0}^\infty \varepsilon_n a_n</math>

converges in <math>X.</math>