Talk:Alternating series test

Untitled

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

The counterexample.

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)

I'm not sure the last sentence is true, and am so marking it. Septentrionalis 04:20, 29 March 2006 (UTC)

'Tis, per example. Septentrionalis 17:55, 29 March 2006 (UTC)

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)

Wasn't very clear

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)

last line

I'm not sure what "this last condition" is meant to be in the last line of text. If it means

   $\left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k-1}\right\vert =a_{k}\!$

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)

I think it is true, because we have in this case $L={\frac {1}{2}},S_{k}={\frac {1}{3}}{\frac {1-({\frac {1}{3}})^{k+1}}{1-{\frac {1}{3}}}}={\frac {1}{3}}{\frac {3}{2}}\left(1-\left({\frac {1}{3}}\right)^{k+1}\right)$, and hence $|L-S_{k}|=|{\frac {1}{2}}-{\frac {1}{2}}\left(1-\left({\frac {1}{3}}\right)^{k+1}\right)|=|\left({\frac {1}{3}}\right)^{k+1}|\leq \left({\frac {1}{3}}\right)^{k}=|S_{k}-S_{k-1}|=a_{k}$ Elmextube (talk) 20:02, 8 December 2009 (UTC)

Index of the sum

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)

Proof of convergence

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)

Proof of partial sum error

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

Merge into alternating series

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)

Confusing proof

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)

New proof with better bound, which agrees with that of alternating series

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)

Corrections of the definition of the alternating series (to allow only positive general terms) so that the error estimate works. Adding citation from one of the standard Calculus text.

For learners of real analysis, the Alternating series page and the Dirichlet's Test are more helpful. — Preceding unsigned comment added by Est nomis (talk • contribs) 19:15, 14 March 2013 (UTC)