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).
(** Coefficients *)

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).