Theorem. If converges then tends to as approaches in such a way that remains bounded.
Here was assuming that and convergence takes place at .
But we let and for example, right?
That is correct. The usual statement of Abel's theorem is that if the series has radius of convergence R, and converges at a point on the circle of convergence (so ), then as nontangentially (meaning that that the angle between and the tangent at is bounded away from 0).