Editing Absolute convergence (section)

===Series with coefficients in more general space===
The term [[unconditional convergence]] is used to refer to a series where any rearrangement of its terms still converges to the same value. For any series with values in a normed abelian group <math>G</math>, as long as <math>G</math> is complete, every series which converges absolutely also converges unconditionally.

Stated more formally:
{{math theorem| Let <math>G</math> be a normed abelian group. Suppose 
<math display="block">\sum_{i=1}^\infty a_i = A \in G, \quad \sum_{i=1}^\infty \|a_i\|<\infty.</math>
If <math>\sigma : \N \to \N</math> is any permutation, then 
<math display="block">\sum_{i=1}^\infty a_{\sigma(i)}=A.</math>}}

For series with more general coefficients, the converse is more complicated. As stated in the previous section, for real-valued and complex-valued series, unconditional convergence always implies absolute convergence. However, in the more general case of a series with values in any normed abelian group <math>G</math>, the converse does not always hold: there can exist series which are not absolutely convergent, yet unconditionally convergent.

For example, in the [[Banach space]] ℓ<sup>∞</sup>, one series which is unconditionally convergent but not absolutely convergent is:
<math display=block>\sum_{n=1}^\infty \tfrac{1}{n} e_n,</math>

where <math>\{e_n\}_{n=1}^{\infty}</math> is an orthonormal basis.  A theorem of [[Aryeh Dvoretzky|A.&nbsp;Dvoretzky]] and [[Claude Ambrose Rogers|C.&nbsp;A.&nbsp;Rogers]] asserts that every infinite-dimensional Banach space has an unconditionally convergent series that is not absolutely convergent.<ref>Dvoretzky, A.; Rogers, C. A. (1950), "Absolute and unconditional convergence in normed linear spaces", Proc. Natl. Acad. Sci. U.S.A. '''36''':192–197.</ref>