since log x is a slowly varying function, you can more or less ignore it in this case

By that I mean the first series will diverge and the second will converge.

NOW, you can either do the direct comp or limit comp tests.

Since the essential p=.5 in the first series you can use any p in (.5,1] say p=2/3.

Compare that series to

or better yet

Next show that and you're done

Showing this limit is infinite means your terms are larger hence your series is larger

and we all know that the harmonic series diverges.