(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope R_scope.
(**********)
(** * Formalization of alternated series *)
Definition tg_alt (Un:nat -> R) (i:nat) : R := (-1) ^ i * Un i.
Definition positivity_seq (Un:nat -> R) : Prop := forall n:nat, 0 <= Un n.
(***************************************)
(** * Application : construction of PI *)
(***************************************)
Definition PI_tg (n:nat) := / INR (2 * n + 1).
(** 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).