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 *)
(** ** Convergence criteria *)
(** D'Alembert criterion for power series *)
(** Equivalence with standard library Reals *)

Lemma CV_radius_Reals_2 (a : nat -> R) (x : R) :
Rbar_lt (Finite (Rabs x)) (CV_radius a)
-> exists r : posreal, CVU (fun n x => sum_f_R0 (fun k => a k * x ^ k) n) (PSeries a) x r.