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What is the best solution for assigning this series for Absolute convergence or Conditional convergence?

I think that we can divide this Summation to two Summations: with positive and with negative sign.

Series: \begin{eqnarray} \sum_{n=1}^{\infty} (-1)^n*{\frac{arcctg(n)}{\sqrt{n}}} \end{eqnarray}

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  • $\begingroup$ "Best" in what sense ? $\endgroup$
    – user65203
    May 5, 2019 at 16:21
  • $\begingroup$ @YvesDaoust I mean the shortest solution, which can be very obviously $\endgroup$
    – Egor
    May 5, 2019 at 16:27
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    $\begingroup$ The series of the absolute values of the terms can be compared to $\sum_n \frac{1}{\sqrt{n}}$. $\endgroup$
    – logarithm
    May 5, 2019 at 16:28
  • $\begingroup$ @logarithm: no, this is inconclusive. $\endgroup$
    – user65203
    May 5, 2019 at 16:47
  • $\begingroup$ @YvesDaoust If you don't know what you are talking about, then learn it. $\endgroup$
    – logarithm
    May 5, 2019 at 16:53

1 Answer 1

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Hint:

$$ {\frac{\text{arccot}(n)}{\sqrt{n}}}\sim\frac1{n^{3/2}}, $$ absolute.

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  • $\begingroup$ I respect your answer, but can you describe it a bit? $\endgroup$
    – Egor
    May 5, 2019 at 16:55
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    $\begingroup$ @EgorRandomize: no, that would make the answer suboptimal. $\endgroup$
    – user65203
    May 5, 2019 at 17:26
  • $\begingroup$ Okay, I understood that you mean that we can divide series to two absolute series. This way, sum of two abs. series is abs. series? $\endgroup$
    – Egor
    May 5, 2019 at 17:29
  • $\begingroup$ @EgorRandomize: there are no signs in the absolute series. $\endgroup$
    – user65203
    May 5, 2019 at 17:30
  • $\begingroup$ Please, describe this transformation, because it was so quickly, that I lost my connection with every math formulas $\endgroup$
    – Egor
    May 5, 2019 at 17:35

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