End PS_incr.

(* Specifications: *)
Lemma PSeries_incr_1 a x :
PSeries (PS_incr_1 a) x = x * PSeries a x.
Lemma PSeries_incr_n (a : nat -> R) (n : nat) (x : R) :
PSeries (PS_incr_n a n) x = x^n * PSeries a x.
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.
Lemma PSeries_decr_1_aux (a : nat -> R) (x : R) :
a O = 0 -> (PSeries a x) = x * PSeries (PS_decr_1 a) x.
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.
Lemma PSeries_decr_n_aux (a : nat -> R) (n : nat) (x : R) :
(forall k : nat, (k < n)%nat -> a k = 0)
-> PSeries a x = x^n * PSeries (PS_decr_n a n) x.
Lemma CV_radius_incr_1 (a : nat -> R) :
CV_radius (PS_incr_1 a) = CV_radius a.
Lemma CV_radius_decr_1 (a : nat -> R) :
CV_radius (PS_decr_1 a) = CV_radius a.