(************************************************************************)
(*         *   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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Require Export Rseries.
Require Export SeqProp.
Require Export Rcomplete.
Require Export PartSum.
Require Export AltSeries.
Require Export Binomial.
Require Export Rsigma.
Require Export Rprod.
Require Export Cauchy_prod.
Require Export Alembert.
Local Open Scope R_scope.

(** Cesaro's theorem *)
Lemma Cesaro :
forall (An Bn:nat -> R) (l:R),
Un_cv Bn l ->
(forall n:nat, 0 < An n) ->
cv_infty (fun n:nat => sum_f_R0 An n) ->
Un_cv (fun n:nat => sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n)
l.