(** * Power series *)
Definition PS_Int (a : nat -> R) (n : nat) : R :=
match n with
| O => 0
| S n => a n / INR (S n)
end.

(* Specifications: *)
Lemma CV_radius_Int (a : nat -> R) :
CV_radius (PS_Int a) = CV_radius a.
Lemma is_RInt_PSeries (a : nat -> R) (x : R) :
Rbar_lt (Rabs x) (CV_radius a)
-> is_RInt (PSeries a) 0 x (PSeries (PS_Int a) x).
Lemma ex_RInt_PSeries (a : nat -> R) (x : R) :
Rbar_lt (Rabs x) (CV_radius a)
-> ex_RInt (PSeries a) 0 x.
Lemma RInt_PSeries (a : nat -> R) (x : R) :
Rbar_lt (Rabs x) (CV_radius a)
-> RInt (PSeries a) 0 x = PSeries (PS_Int a) x.
Lemma is_pseries_RInt (a : nat -> R) :
forall x, Rbar_lt (Rabs x) (CV_radius a)
-> is_pseries (PS_Int a) x (RInt (PSeries a) 0 x).