Definition PS_decr_n (a : nat -> V) (n k : nat) : V := a (n + k)%nat.

Lemma PSeries_decr_n (a : nat -> R) (n : nat) (x : R) : ex_pseries a x -> PSeries a x = sum_f_R0 (fun k => a k * x^k) n + x^(S n) * PSeries (PS_decr_n a (S n)) x.