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The alternating series test requires: - Bn to be decreasing - lim Bn (to infinity) to be 0

In my book I see examples where the series fit the first one, but does not fit the second one and then they use divergence test to prove that it diverges.

But the fact that lim Bn is not 0 already proves this? Since if lim Bn is not 0 the limit of the alternating series will never be 0, the limit of the alternating series will even always be non existent (since it will oscillate)?

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    $\begingroup$ Show an example from your book... $\endgroup$ – imranfat Jul 15 '16 at 16:32
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If you mean that the terms have to converge to $0$ 'anyway': you are right. This is a necessary condition for any series to converge, whether all the terms are positive, negative, alternating or even more complicated. Or the other way around: if the sequence of the terms doesn't converge to $0$, the associated series will definitely diverge.

Coincidentally, there was a recent question that came down to a counter-example for the other condition, namely the monotonically decreasing part:

https://math.stackexchange.com/a/1860421/159845

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The divergence test is the test which says that if the limit does not exist or does not equal zero then the series diverges. So..

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