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$\sum_{n=0}^{\infty}a_n$ converges. It means that the sequence of partial sums is a Cauchy sequence, and it means that $s_n=\sum_{k=n}^{n^2}a_k$ converges against zero.

Does $\sum_{n=0}^{\infty}s_n$ converge? I do not see a reason for it to converge, so I tried to find a good example of $\sum_{n=0}^{\infty}a_n$ such that $\sum_{n=0}^{\infty}s_n$ diverges, but have not so far. Any ideas?

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  • $\begingroup$ Anything works... say $a_n=1/n^2$. Which examples did you check before posting this? $\endgroup$ – Did Nov 24 '15 at 21:04
  • $\begingroup$ @Did a geometric series. $\endgroup$ – Sergey Zykov Nov 24 '15 at 21:05
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    $\begingroup$ Ha. OK. But there are other beasts in the world of series... $\endgroup$ – Did Nov 24 '15 at 21:06
  • $\begingroup$ @Did Thanks for the hint, I tried to expand on it above. $\endgroup$ – Sergey Zykov Nov 24 '15 at 21:22
  • $\begingroup$ @SergeyZykov In your edit, how do you get the $n^4-n^2+1$ factor? You only have $n^2-n+1$ terms. The main issue (that prevents convergence) is that $a_n=1/n^2$ appears many times in $\sum_1^\infty s_n$: it is in $s_{\sqrt{n}}, s_{\sqrt{n}+1, \dots, s_n}$, i.e. roughly $n$ times. $\endgroup$ – Clement C. Nov 24 '15 at 21:25

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