According to worlpramalpha the sum diverges with comparison test...

then what do we compare this sum with, and how to deal with this log function...

by making a few graphs one can see that for example is eventually smaller then this log-function,

which indeed suggests the sum diverges...

Then is an argument by comparing derivatives enough?

Like and

implying the derivative of is eventually smaller then that of the log-function..bla bla

sufficient? More clever ideas ? (a)