Talk:Absolute convergence

message: 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)
thread_title:  Untitled 

message: 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)
thread_title: Large rewrite

message: 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)
thread_title:  p-adic numbers 

message: With regard to the possibility of merging this article with the short article showing absolute convergence implies convergence: the latter article is written at a level accessible to a calculus student, whereas this article is much broader in scope.  To me it seems reasonable to keep both articles and to keep them separate.  However it might be a good idea to link to the calculus article when recalling that completeness and absolute convergence imply convergence. Plclark (talk) 16:39, 9 April 2009 (UTC)
replies: text: If the accessibility of the article is a problem, the right course of action should be to mitigate that problem. I think we shouldn't address accessibility issues by maintaining separate articles. (Yes, this article could certainly use more elementary examples, like alternating series converges but not absolutely, etc. True, this is mentioned but it is mentioned in the way the article assumes readers already know such examples.) -- Taku (talk) 00:20, 10 April 2009 (UTC)

text: Is there anything in the WP:MOS that addresses article accessibility. andyzweb (talk) 02:57, 31 December 2009 (UTC)

text: The more specific WP:MSM has some information on accessibility in mathematical articles. Other than that looking at good and the few featured mathematics articles might show approaches that have worked. --JohnBlackburnewordsdeeds 15:38, 12 January 2010 (UTC)
thread_title:  Merge discussion 

message: Looking to stirr some action on the merge debate I would like to settle it by Jan 21st 2010, at that point I will remove the merge discussion. andyzweb (talk) 05:46, 12 January 2010 (UTC)
thread_title:  reupping merge discussion 

message: This WLOG assumption hides a lot!  This is only valid if the sum of |a_i| converges iff the sum of |Re(a_i)| and sum of |Im(a_i)| converges.  While this may be true, it is disingenuous to say that the proof is complete without it.  The argument I give, expanded from Thm. 3.45 of Rudin in clean and indisputable.  I don't see why the original proof should be kept.
Alsosaid1987 (talk) 21:58, 29 January 2017 (UTC)
thread_title:  Proof that absolute convergence implies convergence 

message: This section is a mess.  Most of the statements made are probably true, as far as my knowledge of this subject goes, and I have taken the liberty of restating and restructuring them somewhat, but as a whole, this section seems to lack focus and require 1) addition of reference(s) and 2) the attention of someone with expertise in integration theory.
-Jimmy Alsosaid1987 (talk) 05:06, 30 January 2017 (UTC)
thread_title:  Absolute integrability 

message: I just added a section on the convergence of sums over sets, largely taken from Tao's analysis book.  However, Tao seems to use a definition of "absolute convergence" which is not the same definition as the one used by other authors for the absolute convergence of an iterated sum.  Tao proves that if a series is absolutely convergent in his sense, then the iterated sum equals the sum over the set of all pairs.  But we don't get a proof that a series is absolutely convergent (in Tao's sense) if and only if the iterated sum is convergent, leaving it open that perhaps the iterated sum could converge even though the sum is not absolutely convergent.  I'm sure that's not true, but I don't know of a reference which establishes the consistency of Tao's definition with that of other authors.  Would love it if someone knew of a reference or otherwise could improve upon what I've written. Addemf (talk) 16:11, 4 August 2021 (UTC)
thread_title:  Convergence of sums over sets 

message: The example of 1 - 1 + 1 - 1 + ... given in the section background does not converge and therefore is not really related to the subject. A better example would be given by the alternating harmonic series. 45.166.163.85 (talk) 19:12, 1 December 2021 (UTC)
thread_title:  1 - 1 + 1 -1 + ... is a bad example for the background discussion