Any chance you could maybe tell us where you got stuck so we don't go over stuff you've got sorted. Just small notes below...
1. as x->0, . Hence so conclude convergence using that etc.
For you could differentiate a few times to get a general formula on how its going to turn out. Should always have in each term so hence each term -> 0.
2. There's a proof of that here Root test - Wikipedia, the free encyclopedia
What are you stuck on?