Comparison test (divergence of sum)

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)