Section PS_minus.
Context {K : AbsRing} {V : NormedModule K}.
Definition PS_minus (a b : nat -> V) (n : nat) : V := plus (a n) (opp (b n)).
End PS_minus.
(** ** Abel's theorem for power series *)
(** * Analysis *)
(** ** Continuity *)
(** ** Differentiability *)
Definition PS_derive (a : nat -> R) (n : nat) :=
INR (S n) * a (S n).
Definition PS_derive_n (n : nat) (a : nat -> R) :=
(fun k => (INR (fact (k + n)%nat) / INR (fact k)) * a (k + n)%nat).

Lemma is_derive_n_PSeries (n : nat) (a : nat -> R) :
forall x, Rbar_lt (Rabs x) (CV_radius a)
-> is_derive_n (PSeries a) n x (PSeries (PS_derive_n n a) x).