Talk:Alternating series test

message: Is the Leibniz test a criterion, I think it isn`t --FHen 09:48, 3 February 2006 (UTC)
thread_title: Untitled

message: The counterexample to the converse (the alternating harmonic series) has nothing to do with the alternating series test.  Rather, it's a counterexample to the converse of the statement "any alternating series that converges absolutely converges," also true, but not the matter at hand.  Since I can't come up with a counterexample, I'm removing it; if someone "in the know" could come up with a better one, that would be great. Twin Bird 17:28, 30 March 2006 (UTC)
replies: text: I'm not sure the last sentence is true, and am so marking it. Septentrionalis 04:20, 29 March 2006 (UTC)
replies: text: 'Tis, per example. Septentrionalis 17:55, 29 March 2006 (UTC)
replies: text: The only counterexamples I can think of are about finite series (a_n = 0 for n above some N). For n < N they can be anything, like n^n :) --11:36, 25 June 2007 (UTC)
thread_title:  The counterexample. 

message: I reworded it a little, in order to say more clearly that if the conditions hold then the series converges.
LDH 19:01, 12 December 2006 (UTC)
thread_title:  Wasn't very clear 

message: I'm not sure what "this last condition" is meant to be in the last line of text. If it means
    \left | S_k - L \right \vert \le \left | S_k - S_{k-1} \right \vert = a_k\!
then this is not true for the example listed  —Preceding unsigned comment added by 58.109.89.67 (talk) 06:01, 3 November 2008 (UTC)
thread_title: last line

message: I changed the starting index of the sums from n=1 to n=0, because previously the proof of convergence used that the sign of every odd-indexed term was positive, which is not true. Also, starting at n=0 is more consistent with alternating series. Elmextube (talk) 20:53, 8 December 2009 (UTC)
thread_title: Index of the sum

message: I made the reference to the monotone convergence theorem explicit. Also one minor change of strict inequality to non-strict inequality.Elmextube (talk) 20:53, 8 December 2009 (UTC)
thread_title: Proof of convergence

message: I added a proof of the partial sum error. Elmextube (talk) 20:53, 8 December 2009 (UTC)
thread_title: Proof of partial sum error

message: I propse we merge this article into alternating series or delete it altogether. The article on alternating series already contains the alternating series test, with a better error bound. MathHisSci (talk) 17:46, 17 November 2010 (UTC)
thread_title:  Merge into alternating series 

message: In the proof for the test, what do the symbols W and S mean? W is defined as the partial sum, but then S suddenly jumps out of nowhere without definition, and it seems like they're both used for partial sums. 
Tebello TheWHAT!!?? 17:54, 24 October 2011 (UTC)
thread_title:  Confusing proof 

message: To Elmextube and others: I have redone both proofs to suit a beginner audience, and improve the error bound to a_{k+1}, which is the same as in alternating series. The convergence proof looks longer but is actually extremely easy to read, and the 3 properties that an is decreasing, nonnegative and converges to 0 are used explicitly in turn.
To Elmextube: I think having an index ranging from 1 to infinity (natural numbers) is better, and it is then easier to talk about odd and even number of terms without confusing students.
To MathHisSci: I think having a separate page for this test has a merit in that many beginning students of calculus will only need the information in this page. a tiny pebble. 07:47, 22 December 2012 (UTC)
thread_title:  New proof with better bound, which agrees with that of alternating series