Explore the absolute convergence of the series with alternating sign [closed]

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

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

Hint:

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

• I respect your answer, but can you describe it a bit?
– Egor
May 5, 2019 at 16:55
• @EgorRandomize: no, that would make the answer suboptimal.
– user65203
May 5, 2019 at 17:26
• 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?
– Egor
May 5, 2019 at 17:29
• @EgorRandomize: there are no signs in the absolute series.
– user65203
May 5, 2019 at 17:30
• Please, describe this transformation, because it was so quickly, that I lost my connection with every math formulas
– Egor
May 5, 2019 at 17:35