Given Cn4^n is convergent as n goes from 0 to infinity:
(i) how is Cn(-2)^n as n goes from 0 to infinity convergent?
(ii) how is Cn(-4)^n as n goes from 0 to infinity not convergent?
The power series always has a "radius of convergence", R, and converges inside that radius- on the interval [-R, R] while diverging outside that interval. Since converges, 4 is either inside that interval or is an endpoint (R= 4).
In either case, since -4< -2< 4, -2 is inside the interval of convergence and converges.
I had first thought that "if converges, then the series must converge but then realized that I was assuming that is positive- and that is not given. If the radius of convergence is 4, knowing that the power series converges at one endpoint of the interval of convergence does not tell us whether the series converges at the other endpoint.
If we know only that converges, then we know that converges but do not know whether converges or not.
If we also know that for all n, then we know that converges (absolutely).