So we have summation of 1/[k(k+1)(k+2)]
We know that 1/[k(k+1)(k+2)] < 1/[(k+1)(k+2)] which I know converges.
Thus, 1/[k(k+1)(k+2)] also converges.
Can someone check my reasoning?
I think this is fine. (Just make sure you have the proof that 1/[(k+1)(k+2)] converges).
As an exercise you can try to formally prove it using your argument above and the definition of convergence. It will give you good experience.
(Also you may want to put these/related questions in the analysis sub-forum.)