Solved. It doesn't converge.
Well, the necessary condition for series to converge is lim a_k -> 0 as k tends to infinity.
In my case, a_k = (k^(j + n))/(m + k)^n = (k^j)*(1 - m/(m + k))^n.
As k -> infinity, (1 - m/(m + k))^n -> 1, while k^j -> infinity.
Which means a_k is unbounded. So, the necessary condition is not satisfied.