Answer
Absolutely Convergent
Work Step by Step
A series $ \Sigma a_n$ is defined as absolutely convergent when the series $ \Sigma |a_n|$ is convergent.
From the series, we notice that
$ \Sigma_{n=1}^{\infty} |\dfrac{\cos ( n \pi) }{n \sqrt n}|= \Sigma_{n=1}^{\infty} \dfrac{1}{n^{3/2}}$
This corresponds to a p-series with common ratio $r=\dfrac{3}{2} \gt 1$.
Remember that a p-series is said to be convergent when the common ratio $ r \gt 1$.
This implies that the series is Absolutely Convergent.