Talk:Convergence

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Two items are said to be convergent if in time they become increasingly like each other. Equally they are divergent if they keep moving apart. However, the word convergent is often used with the second item left implicit.

This isn't really clear enough. There is stuff around on technical convergence, evolutionary convergence by different paths to the same result, and so on.

One question of interest is whether WikiPedia is in broad terms a convergent or divergent phenomenon. Will the number of pages grow without limit (which is possible given arbitrarily long phrases can be used for page names) or will WikiPedia start evolving toward a finished work, or a work where limits exist to changes (with news and topical issues only changing)?

General policy is not to be too navel-gazing about WP on article pages.

Charles Matthews 08:08, 11 May 2004 (UTC)

Contents

[edit] Definition/¨

"Two items are said to be convergent if in time they become increasingly like each other." Well, if both go to infinity, they both diverge. "The difference of two items converge" would be more correct.

The definition of convergence is (seems to be missing):

A sequence \left\{ {a_n } \right\}_{a \ge 1} converges, if there exists a real number a \in \mathbb{R} with the following property:
To any given real value \varepsilon  > 0 there exists a real number N_\varepsilon > 0 with \left| {a_n  - a} \right| < \varepsilon for all n \ge N.


(a is the limit of the sequence \left\{ {a_n } \right\}_{a \ge 1})


[edit] Page structure

I have some qualms about this page's structure. When I was first redirected to it, I was confused by the common-sense description. I think that my confusion resulted from the fact that I could see no introduction or clear structure to the page. I suggest that this article should either become a disambiguation page among informal and formal meanings of the term, or that it should be more clearly structured. I would be willing to make the necessary changes in either case, but I want to know what others think. NatusRoma 06:48, 1 May 2005 (UTC)

[edit] Yup ... should be a disambiguation page

I agree that this should be a disambiguation page, but be sure to include the other senses of the word for "convergence" beside the informal one and the mathematical one in this article. I recall seeing other senses of the word in other specific problem domains, like convergence (biology) or convergence (economics). I gues a Wiki search for "convergence" will get all the senses of the word. ... and thanks for offering to do this so I wouldn't have to Vonkje 20:40, 2 Jun 2005 (UTC)

I agree. Anybody willing to do the work? Oleg Alexandrov 03:39, 6 Jun 2005 (UTC)
I guess I can do it. There are almost 40 articles concerning convergence of which almost half deal with mathematics alone (many of these links are already in this page). I would like to leave this article on Convergence as unchanged as possible. The only exception might be to move the informal description into this discussion page. In the meantime, we can consider merging an improved form of this informal definition into the nicely written article on bargaining (ie: Convergence as an "emergent" property of bargaining).
There are almost 60 articles that link to this Convergence article. I can more relevantly specify some of the more obvious links pertaining to sci-fi and pop culture, but I might do a disservice by attempting the same thing with the mathematics articles. All this might take awhile. Vonkje 18:55, 6 Jun 2005 (UTC)
I have copied and pasted original text from its informal meaning into here:

A common sense example of convergence is in bargaining a price in an informal market. For example, a seller and a buyer may successively make the following offers:

  • Buyer: I'll pay $10.
  • Seller: Impossible! The real value is $100. How about $60?
  • Buyer: Nothing more than $20. That's my final offer.
  • Seller: You're really pushing me. I can't go below $40.
  • Buyer: I'd rather buy in another shop. But if you would accept $30...
  • Seller: OK, $30, it's a deal.
  • Buyer: OK.

Here the sequence of bids and counter-bids evidently converges, quite rapidly, to a common price.

Possibly more to come. Vonkje 20:13, 6 Jun 2005 (UTC)
Okay, ... got all of the senses of Convergence into one page broken out by categories. Links may need some reordering within each category, and correction of typos. What is left is to modify the links shown in the What links here pages. I'm done for the night. Vonkje 00:47, 7 Jun 2005 (UTC) (aka 209.42.38.71).
Hey, you did a lot of great work, thanks! Oleg Alexandrov 00:54, 7 Jun 2005 (UTC)

[edit] Please help me understand something

I am having trouble understanding the following passage:

An infinite series that is divergent does not a priori have any mathematical value. That is, it cannot be used for meaningful computations of its value. Such series are indeed applied: as generating functions, as asymptotic series, or via some summation method.

If divergent series do not a priori have any mathematical value, then why are links listed in the last sentence that each address the topic of divergent series? Is this simply intended as a caveat regarding divergent series? Did the writer intend to use the phrase "lack any mathematical value" instead of "have any mathematical value"? What does "Such series are indeed applied" mean. Wouldn't a discussion or better yet a link to radius of convergence be more helpful? I greatfully look forward to anyone who can shed light on these questions. Vonkje 20:54, 2 Jun 2005 (UTC)

You have a good point. Any suggestions on how to rephrase that passage? Also, please note my removal of some text you inserted, and my comment below. We can talk more about that. Oleg Alexandrov 23:38, 2 Jun 2005 (UTC)
Usually I would have some suggestions, but this one really baffled me! Vonkje 02:02, 3 Jun 2005 (UTC)
Okay, so after re-reading the passage in question:
.. Such series are indeed applied: as generating functions, as asymptotic series, or via some summation method.
I will venture a guess as to what the writer of this passage intended, namely:
.. Nevertheless, attempts have been made to formally characterize divergent series through generating functions, as asymptotic series, or via some summation methods. Vonkje 20:42, 7 Jun 2005 (UTC)

Or how about:

.. Nevertheless, divergent series can be summed formally, as generating functions or asymptotic series, or via some summation method. Oleg Alexandrov 00:23, 8 Jun 2005 (UTC)

Oleg, Your verbage seems to make sense. Since my expertise is not in this particular area (I never studied divergent series) I do not feel confident in replacing the wording, but your wording seems to be a big improvement. Vonkje 22:33, 9 Jun 2005 (UTC)

[edit] On my recent removal

Another reason besides the one stated in the edit summary for removal of those two sentences is the following. The sentence:

The transcendental number Pi provides a classic example of a number that can only be approximated using an infinite series.

is not correct as stated. The infinte series gives the exact value, not an approximation. Also, π can be very easily written in just a finite series, like

π = π+0 +...0

That sentence should have said that

Any partial sum (finite series) containing only rational terms can never be exactly equal to π. As such, π can be only written in an infinite series with rational coefficients, and otherwise one gets only an approximation.

But this would have been too long and complicated paragraph, to insert between two sentences which were making sence together but not separately. Oleg Alexandrov 23:35, 2 Jun 2005 (UTC)

Yes, you are absolutely right. ... I meant to say that "no finite series can effectively compute &pi" (which I think precludes defining &pi in terms of itself as in: π = π+0 +...0). Even so, I'm not so sure if that would have made sense in the spot where I put it. Vonkje 02:02, 3 Jun 2005 (UTC)

[edit] Moving the mathematical terms to their own page

It looks to me that the mathematical meanings of the term take a lot of space on this page. How about making that section much shorter, and instead, move the mathematical meanings to its own page, convergence (mathematics)? Oleg Alexandrov 15:21, 24 Jun 2005 (UTC)

Oleg,
The same thing occured to me when I was reworking this page. I decided against it for two reasons. Firstly, a separate disambiguation page for convergence in mathematics could signal a desire for two Wikipedias, one for power users and the other a ...(ahem)... 'home edition'. ... The second reason is epistemological. I **believe** that all human knowlege derives from mathematics whether we are conscious of this or not. This led me to boldly display the mathematics portion of this disambiguation first.
The risks of this inclusiveness is further co-optation of mathematical concepts by popular culture and its conversion to quasi-mathematical concepts. I can see it now ... a sci-fi thriller titled: "Interval of Convergence", that would have *nothing* to do with infinite series!
Anyway, I would not complain too loudly if someone else were to factor the mathematical meanings out to another page. Vonkje 22:25, 24 Jun 2005 (UTC)
I see. I don't plan to change it, but I will not complain either of somebody else does. Oleg Alexandrov 03:40, 25 Jun 2005 (UTC)
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