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

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.