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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
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803:{\displaystyle \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}})\approx 0.6048986434....}
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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
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862:"An alternating series converges if the terms an converge to 0 monotonically." Below this, the Leibniz rule is stated. I think the same thing is said twice, making the first one redundant.
819:. 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.
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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.
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The error estimate will be incorrect if the general terms are allowed to be only nonnegative. I'm changing it back to strictly positive.
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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.
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848:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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338:= 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
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Anyone care to add a short proof of the
Leibniz Test? I think it would add a lot to the page.
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this article has a lot of room for improvement, take a look at any elementary calculus text.--
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A new proof is given at the "Alternating Series Test" page, which cleans up the presentation.
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Last edited at 21:21, 9 May 2009 (UTC). Substituted at 01:45, 5 May 2016 (UTC)
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Another valid example of alternating series is the following
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