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 *)

Lemma CV_disk_DAlembert_aux (a : nat -> R) (x k : R) :
x <> 0 -> (forall n, a n <> 0) ->
(is_lim_seq (fun n => Rabs (a (S n) / a n)) k
<-> is_lim_seq (fun n => Rabs ((a (S n) * x^(S n)) / (a n * x ^ n))) (Rabs x * k)).