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).