Section PS_incr.
Context {K : AbsRing} {V : NormedModule K}.
Definition PS_incr_1 (a : nat -> V) (n : nat) : V :=
match n with
| 0 => zero
| S n => a n
end.
Fixpoint PS_incr_n (a : nat -> V) (n k : nat) : V :=
match n with
| O => a k
| S n => PS_incr_1 (PS_incr_n a n) k
end.
Definition PS_decr_1 (a : nat -> V) (n : nat) : V := a (S n).
Definition PS_decr_n (a : nat -> V) (n k : nat) : V := a (n + k)%nat.
End PS_incr.
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.