University Calculus: Early Transcendentals (3rd Edition)

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.