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I went through many related posts here on stackexchange but I am not able to find answer to my question. Please don't mark this as a duplicate.

Let $\sum\limits_{n=1}^{\infty}a_n$ be a conditionally convergent series. That is, $\sum\limits_{n=1}^{\infty}a_n$ converges but $\sum\limits_{n=1}^{\infty}|a_n|$ doesn't. I want to show that in this case a subseries $\sum\limits_{n=1}^{\infty} a_{f(n)}$, $f:\mathbb{N}\to\mathbb{N}$ exists that diverges. This I need to prove that a conditionally convergent series can be rearranged to diverge. However, I didn't get any clear proof of this or couldn't prove it myself.

Zeekless
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Peaceful
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    It seems that either the subseries of all positive elements, or the subseries of all negative elements, or both, will diverge. Otherwise $\sum |a_n|$ would be a sum of two converging series. – Zeekless Apr 25 '21 at 07:37
  • Does this answer your question? https://math.stackexchange.com/q/370932/42969 – Martin R Apr 25 '21 at 07:39
  • Your question is not clear enough. You speak of a subsequence that diverges. This may not be the case for a conditionally converging series. And in the following sentence of rearranging the terms of the series. Can you fix that? – mathcounterexamples.net Apr 25 '21 at 07:40
  • @mathcounterexamples.net : I looked at the Wikipedia page of Riemann's rearrangement theorem. The proof that a rearrangement exists so that the series diverges uses the fact that there exists an divergent subseries. What am I missing? – Peaceful Apr 25 '21 at 11:12
  • @MartinR: Not really. The answer there assumes that $\sum a_n$ diverges. On what basis? I know that $\sum |a_n|$ diverges but not $\sum a_n$. That's why we call the series conditionally convergent. – Peaceful Apr 25 '21 at 11:28
  • @Zeekless: I think that answers my question partially. The problem is that we must be able to prove that both these subseries diverge because there exist rearrangments making the original series diverge to $-\infty$ or $\infty$. – Peaceful Apr 25 '21 at 11:50

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