Definition PS_derive_n (n : nat) (a : nat -> R) :=
(fun k => (INR (fact (k + n)%nat) / INR (fact k)) * a (k + n)%nat).

(* Specifications: *)
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).
Lemma ex_derive_n_PSeries (n : nat) (a : nat -> R) (x : R) :
Rbar_lt (Finite (Rabs x)) (CV_radius a)
-> ex_derive_n (PSeries a) n x.
Lemma Derive_n_PSeries (n : nat) (a : nat -> R) (x : R) :
Rbar_lt (Finite (Rabs x)) (CV_radius a)
-> Derive_n (PSeries a) n x = PSeries (PS_derive_n n a) x.
Lemma CV_radius_derive_n (n : nat) (a : nat -> R) :
CV_radius (PS_derive_n n a) = CV_radius a.