Since the series does not terminate, the coefficients as product of many numbers differ by 1 must achieve rapid growth. That means if x is not bounded by 1, the terms may and must diverges eventually. Also, we can refer to ratio test that a series converges at least the absolute ratio is less than 1, and only |x|<1 does work for the condition. The negative power cases can be treated as sum of Geometric series, and it is indeed a category of hypergeometric series.