Definition PS_mult (a b : nat -> R) n :=
sum_f_R0 (fun k => a k * b (n - k)%nat) n.

(* Specifications: *)
Lemma is_pseries_mult (a b : nat -> R) (x la lb : R) :
is_pseries a x la -> is_pseries b x lb
-> Rbar_lt (Rabs x) (CV_radius a) -> Rbar_lt (Rabs x) (CV_radius b)
-> is_pseries (PS_mult a b) x (la * lb).
Lemma ex_pseries_mult (a b : nat -> R) (x : R) :
Rbar_lt (Rabs x) (CV_radius a) -> Rbar_lt (Rabs x) (CV_radius b)
-> ex_pseries (PS_mult a b) x.
Lemma PSeries_mult (a b : nat -> R) (x : R) :
Rbar_lt (Rabs x) (CV_radius a) -> Rbar_lt (Rabs x) (CV_radius b)
-> PSeries (PS_mult a b) x = PSeries a x * PSeries b x.