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I have an alternating series $\displaystyle\sum_{n=0}^{\infty} (-1)^{n+1}a_n.$ I know that it is convergent. I have to find $a_n$ which is non-decreasing.

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closed as unclear what you're asking by This is much healthier., Sharkos, Rick Decker, Michael Albanese, rogerl Jul 16 at 0:54

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

From your title it appears that you're looking for a subsequence but it's not clear from your text. The usual way to write a subsequence is ${a_{n_k}}$ ($k = 1, 2, 3...$). As stated, I think the claim that there's such a subsequence which you're trying to prove is false: suppose $a_n = 1/n$, then the series is convergent but every subsequence of ${a_n}$ is decreasing. –  Jonah Sinick Oct 29 '12 at 1:09
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1 Answer 1

If the series is convergent, then $a_n\to 0$, and as $a_n\ge 0$, cannot have an infinite increasing subsequent. And a nondecreasing $a_n$ then must be quasiconstant $0$.

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