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

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).