Talk:Alternating series

$$a_n$$ positive versus nonnegative
Many references (for example http://mathworld.wolfram.com/AlternatingSeries.html ) require that the definition of alternating series require $$a_n>0$$ for all n or $$a_n < 0$$ for all n.

Wrong statement
The sequence of positive a_n's must be monotone decreasing after a certain point in order for the series to converge. Even then, the Liebniz's test is not an if and only if. For the first case, notice that $$a_n = 1/n$$ if n is odd and $$a_n$$ = 1/2^n if n is odd can be made to a divergent alternating series even though the limit as n tends to infinity of $$a_n$$ is 0.

proof of Leibniz test
Anyone care to add a short proof of the Leibniz Test? I think it would add a lot to the page. Lavaka 05:45, 21 September 2006 (UTC)

I added a proof the way it came to my mind. In some texts it is a little more convoluted, although I think in the form stated here it is easy to see the idea of what's going on. Tinchote 5:17, 30 August 2007 (UTC)

Not sure about alternating sequence

 * $$\sum_{n=1}^\infty \frac{(-1)^n+1}{n},$$

This produces a term of zero for every time n is odd and 2/n every time n is even making this sequence the same as the harmonic series.

Is it not supposed to be this sequence instead?


 * $$\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}.$$

Also, the proof of the Alternating Series Test has the subscripts switched, namely n + 1 and m. Also it is not explicitly stated how Sn and Sm are necessarily positive, which is necessary for the Cauchy criterion to apply at all, since even though Sm - Sn < an+1, if Sm - Sn is not bounded below by zero, the statement does not imply convergence.

—Preceding unsigned comment added by Earthstrike (talk • contribs) 23:15, 18 August 2009 (UTC)

New proof of the Alternating Series Test
A new proof is given at the "Alternating Series Test" page, which cleans up the presentation.

The error estimate will be incorrect if the general terms are allowed to be only nonnegative. I'm changing it back to strictly positive.

a tiny pebble. 18:52, 14 March 2013 (UTC)

Example with zeta
The following example was given:
 * Another valid example of alternating series is the following

\sum_{k=0}^{\infty} \frac{(-1)^{k}}{\sqrt{k+1}}=1-\frac{1}{\sqrt{2}} +\frac{1}{\sqrt{3}} -\frac{1}{\sqrt{4}} +\frac{1}{\sqrt{5}} \cdots=-(\sqrt{2} -1)\zeta(\frac{1}{2})\approx0.6048986434.... $$ It may be that the zeta is the Riemann zeta function. As a sophisticated example with no reference, it has been removed. Should someone have a demonstration or reference, it may be added to the section of examples of alternating series.Rgdboer (talk) 03:01, 16 December 2014 (UTC)

Assessment comment
Substituted at 01:45, 5 May 2016 (UTC)