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We need to show that monotonicity in the alternating series test is necessary. I know that it is otherwise, examples like the following:

Monotonicity in alternating Series

could be constructed. But how can we rigorously prove this?

  • It’s not necessary, though it is sufficient. For example you can break monotonocoty by adjusting the first 20 terms – Calvin Khor Nov 16 '20 at 00:11
  • They mean that monotonicity cannot be dropped from the theorem. Not that convergence of an alternating series implies monotnicity. – Kavi Rama Murthy Nov 16 '20 at 00:12
  • @KaviRamaMurthy i think you are right, but it certainly sounds very odd to me; of course its certainly not true that monotonicity is necessary and sufficient for the alternating series to converge... – Calvin Khor Nov 16 '20 at 02:48

1 Answers1

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To prove monotonicity is necessary, you simply need a counterexample, i.e. a divergent alternating series which has $|a_n|\to 0$, but not monotonically. Try: $$\frac{1}{\sqrt 2-1}-\frac{1}{\sqrt 2+1}+\frac{1}{\sqrt 3-1}-\frac{1}{\sqrt 3+1}\ldots$$

Rhys Hughes
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  • How come we can just show a counter-example? When we prove something don't we need to rigorously prove it? –  Nov 16 '20 at 00:23
  • You need to understand that your task is to disprove the claim: "Every series where the terms alternate signs and converge to 0 converges."

    To disprove a claim of the form "Every something X has property Y" it is enough to provide one instance of X that does not have property Y. If the demonstration that the given instance of X does indeed not have property Y is given to satisfactory levels, that's all there is to do.

     – Ingix Nov 16 '20 at 13:11
  • @user1234 You know that having the properties of $|a_n|\to 0$ and monotonicity guarantees a given series is convergent. Your question is asking whether monotonicity is necessary, that is, is just $|a_n|\to 0$ sufficient to determine that the series is convergent, or do we need monotonicity to guarantee it? The example I give in my answer proves that $|a_n|\to 0$ without monotonicity is not sufficient to guarantee convergence. – Rhys Hughes Nov 16 '20 at 16:53