A handy test in this area is the n-th root test: given a series , for which the sequence tends to a limit , the series is convergent if and divergent if . (In the case , the test gives no information.)
Applied to your series (1), we see that and it isn't too hard to show that . Hence the limit and the series is convergent for any value of x.
For your series (2) a useful trick is to regard it as a special value of a function , evaluated at . We start with and differentiate to get . Multiply by x and differentiate again to find f.