Talk:Absolute convergence
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Why is products of series here? —The preceding unsigned comment was added by 129.7.250.116 (talk • contribs) 14:21, 31 August 2006 (UTC)
- It's not absolutely, logically necessary, but it demonstrates an important application of absolute convergence. Melchoir 14:54, 31 August 2006 (UTC)
Another user
The use of the word "theorem" in the (nice) discussion that starts: ". . . this theorem can be imagined as. . ." is confusing. There is a theorem that absolute convergence implies convergence: Is that meant here? This implies the too-loose-to-be-a-theorem vebiage about "rearrangements...intuitive." But the reader has not yet met the theorem, so it is premature to call it such here.
I generally like giving intuitive grounding for math, and I think this section does a good job of doing so. I can't think of good substitute for "theorem". Might be necessary to add more words.
[edit] Large rewrite
When I read this page I found that it didn't give enough information about the relation between absolute convergence and convergence, and especially that it conflated absolute and unconditional convergence. So I rewrote quite a bit of it.
The rewrite admittedly has a more technical feel to it: it seemed to me that to get to the bottom of absolute convergence, one should see it in proper generality, whence a section on absolute convergence for series with values in a normed abelian group. The definition of normed abelian group I use comes from Kaplansky's Set Theory and Metric Spaces (and should probably be referenced). (Apparently some people use this terminology without the symmetry axiom (iii), which means the norm might not induce a metric...) So far as I know, the main case of interest is that of a Banach space, but it seems satisfying to observe that the proofs of the easier theorems work verbatim in this level of generality. I also feel that knowing that unconditional and absolute convergence is not the same thing in general because the two fail to coincide on every infinite dimensional Banach space is an interesting and enlightening fact even if you don't really know or care much about Banach spaces (as I do not).
Probably a more careful structuring between the sections "A more general setting for absolute convergence" and "Relations with convergence" would make the latter section more palatable for students at the honors calculus / undergraduate real analysis level who do not know what a Banach space is. Perhaps someone has good ideas for this?
I started out the rewrite carrying along the parallel between series and integrals but found it tiresome, so currently there's just a short section at the bottom acknowledging that the concept makes sense for integrals. Notice that the relationship between absolute integrability and integrability depends strongly on what integration theory you're using: for the proper Riemann integral one has integrability of f implies integrability of |f| but not conversely; for the Lebesgue integral the two are equivalent by definition; for the Kurzweil-Henstock integral I think the implication goes the other way. Maybe it's better to have this as a separate article? Plclark (talk) 15:48, 29 December 2007 (UTC)Plclark
[edit] p-adic numbers
For the field of p-adic numbers, I think the following holds: Any convergent series is unconditionally convergent (though not necessarily absolutely convergent). This is because a series converges iff its terms are a zero series, and this condition remains unchanged under permutation of the terms.
On the other hand, there are quite "natural" convergent series which do not absolutely converge: e.g. the sequence 1, p,..., p, p^2, ..., p^2, p^3, ... (taking each summand p^n p^n times).
Thus, the term "unconditionally convergent" is the more "natural" in this case, IMO. --Roentgenium111 (talk) 22:27, 14 April 2008 (UTC)
