Section ConvergenceCircle.
Context {K : AbsRing} {V : NormedModule K}.
End ConvergenceCircle.
(** Convergence disk *)
Definition CV_disk (a : nat -> R) (r : R) :=
ex_series (fun n => Rabs (a n * r^n)).
Definition CV_radius (a : nat -> R) : Rbar :=
Lub_Rbar (CV_disk a).
(** Convergence theorems *)

Lemma CV_disk_inside (a : nat -> R) (x : R) :
Rbar_lt (Finite (Rabs x)) (CV_radius a)
-> ex_series (fun n => Rabs (a n * x ^ n)).