Explore the absolute convergence of the series with alternating sign

Question

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}

Answer 1

Hint:

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