Answer
Absolutely Convergent
Work Step by Step
A series $ \Sigma a_n$ is said to be absolutely convergent when $ \Sigma |a_n|$ is convergent.
We notice that $ \Sigma_{n=1}^{\infty} |\dfrac{\cos ( n \pi) }{n \sqrt n}|= \Sigma_{n=1}^{\infty} \dfrac{1}{n^{3/2}}$
We see a p-series with common ratio $r=1.5 \gt 1$; when the common ratio $r \gt 1$, then a p-series is convergent.
Therefore, the given series is Absolutely Convergent.
