Talk:Convergence tests

For the ratio test: If all a(n) are positive then why absolute value is needed within the lim? That is, we should omit the positive condition, and let the test extend to the alternating series, such as a(n+1) = -a(n)/2.

How about the alternating series test? Fishie2610 (talk) 00:05, 12 June 2009 (UTC)

If i'm not mistaken Leibniz also have an important test regarding convergence which says that if An->0 and An>An+1 then the sum of An*(-1^n) should converge. —Preceding unsigned comment added by 84.229.138.185 (talk) 15:38, 29 July 2010 (UTC)

limit of the summand

the article says if the limit of the summand is zero the sequence of partial sums if cauchy. I'm not sure of this, but AFAIK if a sequence is cauchy in the reals it converges, but that doesnt happen to the harmonic series, even if it's summand tends to zero. — Preceding unsigned comment added by 186.18.76.220 (talk) 22:58, 15 November 2011 (UTC)

It says so here http://en.wikipedia.org/wiki/Convergent_series#Cauchy_convergence_criterion — Preceding unsigned comment added by 186.18.76.220 (talk) 23:01, 15 November 2011 (UTC)

I can't resolve this, but it ought to be resolved. Both of the above comments seem to be correct. In addition, is it reallyl true that if the limit is undefined then the sum diverges? No source is given. --editeur24 (talk) 05:05, 18 December 2020 (UTC)

I see the resolution now. As the section says, if the summand tends to zero, the test is inconclusive. That's what happens when you apply it to the harmonic series, where the summand tends to zero. So to converge, the sequence must be Cauchy, but even if it's Cauchy, it might not converge. A necessary but not sufficient condition for convergence.

On my second point: this article should really have a few sentences in the intro on what convergence means, as well as its link to the main convergence article. Not an entire section-- this is about tests, not the definitions of convergence-- but a little bit of intuition. --editeur24 (talk) 16:35, 18 December 2020 (UTC)

terms must be positive or not

this is a huge discrepancy between this page and the test descriptions on the convergent series page. Someone please fix / clarify. — Preceding unsigned comment added by 75.186.86.53 (talk) 11:32, 18 March 2016 (UTC)

The polishing of the statement of the Leibniz criterion

The language used to state the criterion is rather verbose. I suggest that we apply the bullet point style for the statement, like the other criteria on the page. 13:27, 4 January 2017 (UTC)~ Appleuseryu (talk) 13:27, 4 January 2017 (UTC)

Should the Absolute convergence be called a test and have its own section?

Absolute convergence test

Every absolutely convergent series converges.

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Should this be its own section, as it is presently, and is it really a "test"? It is a useful remark-- should it be put elsewhere?

--editeur24 (talk) 05:10, 18 December 2020 (UTC)

Should this article include examples? (discussion of reverted changes)

I added examples to the article. JayBeeEll reverted them. He said,

All of these examples have errors (the universal one being the confusion of a series and the associated sequence of summands), but more broadly I question the value of examples in an article like this -- every section is a direction to the associated main article

JayBeeEll has two points. First, errors. In a couple of examples I call a sequence a series. That can be fixed. I imagine any other errors can too.

As JayBeeEll says, though, the main issue is his second one: whether examples belong in this article. I think they do, because examples convey information very well, often better than the definitions. Examples are very common in Wikipedia math articles for this reason. Rather than working through the definition and notation, it is often easier just to look at the example and say "Ah, that's what's going on with this technique". Also, examples show why there is more than one test, because some series require one test, some another.

I see also that the article does have an Examples section, though just with one example, for the Cauchy condensation test. I think that example would be more useful if put in the Cauchy condensation test section. Also the section on the root test has an example, though it is not explained. Should those two examples be deleted?

Any comment on whether I should look at my examples again, fix whatever errors I find, and add them back? editeur24 (talk) 22:29, 21 December 2020 (UTC)

Thanks for your message. I'm sorry no one else has weighed in yet (maybe because of the time of year?) because I think it would be nice to get more views. First, points of agreement: (1) examples are, in general, more enlightening than definitions; (2) the Examples section as currently constructed is bizarre and it would make more sense if that lone example were in the relevant section; and (3) I agree that any errors in the examples you gave could be fixed.

Second, let me lay out why I don't think lots of examples are a good idea in this article. At present, this article is constructed as a navigational aid, essentially: it presents a minimum of useful information about each test it covers, but it routes the reader to the main article on all of the tests, where more information can be found. In my opinion, choosing good examples for the various tests is quite challenging: for example, I do not think the geometric series is a good illustration for the ratio test because the whole spirit of the ratio test is something like "I understand geometric series; let me identify series that behave sufficiently like geometric series that I can understand them". To illustrate the power of the method you would want something where the sum is not actually easy to compute, like $\sum {\frac {\sqrt {n}}{2^{n}}}$ or whatever. But then to make the example meaningful, it is necessary to include a demonstration of the working, and all of a sudden we have a huge chunk of material on each and every test, duplicating what should be in an Examples subsection at each of the linked articles and making this page harder to navigate and maintain.

I hope that other editors will also give their thoughts on these questions. --JBL (talk) 17:15, 26 December 2020 (UTC)