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

Lemma mk_pseries (f : R -> R) (M : R) (r : Rbar) : (forall n x, Rbar_lt (Finite (Rabs x)) r -> (ex_derive_n f n x) /\ Rabs (Derive_n f n x) <= M) -> forall x, Rbar_lt (Finite (Rabs x)) r -> is_pseries (fun n => Derive_n f n 0 / INR (fact n)) x (f x).