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

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).