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

Lemma PSeries_ext_recip (a b : nat -> R) (n : nat) : Rbar_lt (Finite 0) (CV_radius a) -> Rbar_lt (Finite 0) (CV_radius b) -> locally 0 (fun x => PSeries a x = PSeries b x) -> a n = b n.