(** Definition of cosinus *)
Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).

(* Specifications: *)
Lemma simpl_sin_n :
forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)).
Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0.
Lemma sin_no_R0 : forall n:nat, sin_n n <> 0.
Definition sin_in (x l:R) : Prop :=
infinite_sum (fun i:nat => sin_n i * x ^ i) l.