According to worlpramalpha the sum $\displaystyle \sum_{k=1}^{\infty}k\log(1+\frac{1}{k(k+2)})$ 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 $\displaystyle 1/k^2$ is eventually smaller then this log-function,

which indeed suggests the sum diverges...

Then is an argument by comparing derivatives enough?

Like $\displaystyle \frac{d}{dx}1/x^2 = -1/x^3$ and $\displaystyle \frac{d}{dx}\log(1+\frac{1}{x(x+2)}) = \frac{-2}{x(x^2+3x+2)}$

implying the derivative of $\displaystyle 1/x^2$ is eventually smaller then that of the log-function..bla bla

sufficient? More clever ideas ? (a)