For the series:

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{(n)^(1/2)(logn)^3}$

To show it is divergent/convergent would you first apply the Cauchy test:

$\displaystyle 2^n \frac{1}{(2^(n/2) (nlog(2)^)3}$

$\displaystyle \frac{2^(n/2)}{(n^3 log(2)^3)}$

Then apply the ratio test to get the lim = sqrt(2) so it is a divergent series?

Thanks.