(** Now, PI is defined *)
Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).

(** We can get an approximation of PI with the following inequality *)
Lemma Alt_PI_ineq :
forall N:nat,
sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <=
sum_f_R0 (tg_alt PI_tg) (2 * N).