Is that really k! and not n!? If so, the sum can be written as and the radius of convergence is just the radius of convergence of . As Prove It says, the simplest way to find radii of convergence for power series is to use the ratio test:
If , then and . Since the ratio test says that will converge uniformly for the limit of the ratio less than one, we must have |x|< 1 and the radius of convergence is 1.
Now, the boundary points are x= -1 and x= 1. If x= 1, that is k(1+ 1+ 1+ ...). Does that converge? At x= -1, it is k(1- 1+ 1- 1+ ....). Does that converge?
If it is actually it's a little different:
and you must take the limit, as n goes to infinity, of that.
2)
Post it has already told you that the radius of convergence is 4. That means that the endpoints of the interval of convergence are x= -4 and x= 4.
For x= 4, that series is or
That's not going to be easy to decide!
For x= -4 that series is . Because that is an 'alternating' series, it will converge as long as the individual terms are eventually decreasing.