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?