Talk:Comparison test

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Mathematics rating:	Start Class	Mid Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Upgraded to mid priority. This articles needs some examples, better references, a discussion of the relation with other convergence tests, an explanation of the comparison test for improper integrals, and so forth. Jim 00:07, 5 September 2007 (UTC)

"if a_n <= b_n for all n, and suppose that sum (from n= 0 to infinty) b_n is convergent. Then sum a_n is convergent."
(Craw, 2002)

Notice "for all n". Wiki says that "for sufficiently large n".

Also, please explain why comparing the ratios of a series to another known series is betten than comparing the ratio to 1, as in the d'Alembert test. I'm sure there must be a reason, but I can't figure it.

Craw, I., 2002, The Comparison Test, The University of Aberdeen, Available from http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node50.html

212.159.75.167 17:33, 10 December 2006 (UTC)Timbo

[edit] Comparison test of the second kind

In my opinion the described comparison test of the second kind is plain wrong. As a counterexample you can take a series Failed to parse (Cannot write to or create math output directory): b_n=s^n

with s < 1 and 

a series a_n = rⁿ with r > 1 (i.e. a converging and a non-converging geometric series) and obviously the condition is fulfilled with C = r / s, but as obivously a_n is not converging while b_n is. This could be fixed by requiring C < = 1, but then I could directly use the ratio test (at least, I can't see a point in comparing to another series in that case). So, I would vote for removal of this passage and any references to it. 134.169.77.186 10:29, 29 August 2007 (UTC) (ezander)

The material on the "comparison test of the second kind" was completely incorrect, and has now been deleted. Jim 00:03, 5 September 2007 (UTC)

[edit] Name of Test

The test is also taught as the "direct comparison test", to separate it from the "limit comparison test", at least at my university. —Preceding unsigned comment added by ThomasOwens (talk • contribs) 16:15, 21 October 2007 (UTC)