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 (a : nat -> R) (x:R) l :
(forall n:nat, a n <> 0) ->
is_lim_seq (fun n:nat => Rabs (a (S n) / a n)) (Finite l) ->
(l = 0 \/ (l <> 0 /\ Rabs x < / l))
-> CV_disk a x.