Talk:Integral test for convergence

Non integrable function?
What if the function f(x) mentioned in the theorem is not integrable? Some books add the hipothesis that f(x) is continuous, but here it is not assumed, then what guarantees that it is integrable? — Preceding unsigned comment added by 186.18.76.220 (talk) 20:33, 23 November 2011 (UTC)


 * "non-negative monotone decreasing function" is enough to use upper and lower Riemann sums to show it is integrable --Rumping (talk) 07:04, 24 September 2012 (UTC)

The Integral test makes a "bridge" between two important chapters of the mathematical basis - Integral Calculus and Series. It can be stated under the single hypothesis of monotony! See http://www.youtube.com/watch?v=2pvuCEnb60k José salazar, ISCAL 2013 — Preceding unsigned comment added by 79.168.141.210 (talk) 16:08, 30 March 2014 (UTC)

Removed section
I have removed the following

Other Statement
The criterion can be stated under the single hypothesis of monotony. Full details at www.youtube.com/watch?v=2pvuCEnb60k ISCAL, José Salazar - 2011 since it looks like WP:OR to me. Also, YouTube is not a reliable secondary source and this would require such a source to remain in the article. Bill Cherowitzo (talk) 03:48, 4 April 2014 (UTC)
 * Positivity is not needed because monotonic functions always have a limit. If the limit is null, the decreasing function is necessarily positive. If the limit is not null then fail the necessary conditions of convergence of an improper integral and a series.
 * Many authors state for Integral Test the continuity requirement, may be as a integrability condition. In the proof above there is no reference about this. The fact is that the monotonic functions on [a,+∞[ are monotonic and bounded in any bounded interval and therefore are also integrable. Thus, if we take the monotony in the statement, we do not need to require continuity.
 * Once the series and improper integrals with f or -f are of the same nature (convergent or divergent), we can conclude that the criterion can be stated under the single condition of monotony.

About Removed Section
I mentioned my video on YouTube because I have not had time to write it all here. In the video there are all sources, including the oldest publication, that I Know, with the proof from the one condition hypothesis - "Introduction to Mathematical Analisys", 1987, from a very important portuguese matematician,  Jaime Campos Ferreira. In the Reference section I had added this referente, but I am sure that I was not clear enough. José Salazar. — Preceding unsigned comment added by Jmegsalazar (talk • contribs) 15:24, 8 April 2014 (UTC)
 * I didn't mean to sound harsh when I pulled this section out of the article, but Wikipedia has a very broad definition of "original research" (OR) and does not allow it. Work that we do ourselves, even detailed proofs which can be checked by anyone, is considered OR unless it already appears in a reliable secondary source (and we are merely reporting on what is in the source.) As a check on editors, Wikipedia has a requirement of verifiability (WP:V), that is, making sure that the statements have an appropriate source. The problem I have with your contribution has nothing to do with whether or not I think it is correct (or even significant), but rather with the fact that I can not verify it. You have not provided the information needed to do this. As you say in your YouTube video, you only have one source written by your teacher Ferreira. This appears, I assume, as an article(?) in "Introduction to Mathematical Analysis" written in Portuguese(?). I can not tell if this is a primary or secondary source, or whether it was referred or not (I am assuming it is an article since the volume had an editor) which is a measure of reliability. The best source would be something (preferably in English, but that is not a requirement) which talks about Ferreira's contribution, but at a minimum you should provide the needed bibliographic data on the source that you have. As an aside, you would not need to expand the section by very much since this is an encyclopedia entry and not an exposition of the material. Bill Cherowitzo (talk) 16:58, 8 April 2014 (UTC)

Incorrect graph?
The graph on the top right corner of the page doesn't illustrate what it's supposed to. It shows the function y=1/x, not the harmonic series which is y=1/1+1/2+1/3+...+1/x, which is clearly different, as it diverges (always keeps climbing), while 1/x approaches 0 as x approaches positive infinity (i.e. it converges to 0). 206.248.158.23 (talk) 03:42, 22 November 2015 (UTC)
 * No, the graph is correct - it is a graph of 1/x. The point is that the nth partial sum of the harmonic series is bounded below by ln(n) (the antiderivative of the graphed function), which is unbounded as n → infinity, and hence that sequence must diverge.--Jasper Deng (talk) 06:48, 22 November 2015 (UTC)