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.

Lemma PSeries_decr_1 (a : nat -> R) (x : R) :
ex_pseries a x -> PSeries a x = a O + x * PSeries (PS_decr_1 a) x.