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(*         *   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)         *)
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Local Open Scope R_scope.
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(** * 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.
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(** * Application : construction of PI *)
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Definition PI_tg (n:nat) := / INR (2 * n + 1).
(** Now, PI is defined *)
Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).

Lemma Alt_PI_RGT_0 : 0 < Alt_PI.