i have to determine the radius of convergence of the following power series, state the regions which the series converge uniformly and study the convergence at the boundary of the interval of convergence
1)
2)
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.