(** Sums on even and odd *)

(* Specifications: *)
Lemma is_pseries_odd_even (a : nat -> R) (x l1 l2 : R) :
is_pseries (fun n => a (2*n)%nat) (x^2) l1 -> is_pseries (fun n => a (2*n+1)%nat) (x^2) l2
-> is_pseries a x (l1 + x * l2).
Lemma ex_pseries_odd_even (a : nat -> R) (x : R) :
ex_pseries (fun n => a (2*n)%nat) (x^2) -> ex_pseries (fun n => a (2*n+1)%nat) (x^2)
-> ex_pseries a x.
Lemma PSeries_odd_even (a : nat -> R) (x : R) :
ex_pseries (fun n => a (2*n)%nat) (x^2) -> ex_pseries (fun n => a (2*n+1)%nat) (x^2)
-> PSeries a x = PSeries (fun n => a (2*n)%nat) (x^2) + x * PSeries (fun n => a (2*n+1)%nat) (x^2).
Lemma PSeries_const_0 : forall x, PSeries (fun _ => 0) x = 0.
Lemma CV_radius_const_0 : CV_radius (fun _ => 0) = p_infty.
Section PS_opp.