Library Stdlib.Reals.Alembert

Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import SeqProp.
Require Import PartSum.
Require Import Lra.

Local Open Scope R_scope.


Lemma Alembert_C1 :
  forall An:nat -> R,
    (forall n:nat, 0 < An n) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.

Lemma Alembert_C2_aux_positivity :
  forall Xn : nat -> R,
    let Yn i := (2 * Rabs (Xn i) + Xn i) / 2 in
    (forall n, Xn n <> 0) ->
    forall n, 0 < Yn n.

Lemma Alembert_C2_aux_Un_cv :
  forall Xn : nat -> R,
    let Yn i := (2 * Rabs (Xn i) + Xn i) / 2 in
    (forall n, Xn n <> 0) ->
    Un_cv (fun n:nat => Rabs (Xn (S n) / Xn n)) 0 ->
    Un_cv (fun n => Rabs (Yn (S n) / Yn n)) 0.

Lemma Alembert_C2 :
  forall An:nat -> R,
    (forall n:nat, An n <> 0) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.

Lemma AlembertC3_step1 :
  forall (An:nat -> R) (x:R),
    x <> 0 ->
    (forall n:nat, An n <> 0) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
    { l:R | Pser An x l }.

Lemma AlembertC3_step2 :
  forall (An:nat -> R) (x:R), x = 0 -> { l:R | Pser An x l }.

Theorem Alembert_C3 :
  forall (An:nat -> R) (x:R),
    (forall n:nat, An n <> 0) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
    { l:R | Pser An x l }.

Lemma Alembert_C4 :
  forall (An:nat -> R) (k:R),
    0 <= k < 1 ->
    (forall n:nat, 0 < An n) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.

Lemma Alembert_C5 :
  forall (An:nat -> R) (k:R),
    0 <= k < 1 ->
    (forall n:nat, An n <> 0) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.

Lemma Alembert_C6 :
  forall (An:nat -> R) (x k:R),
    0 < k ->
    (forall n:nat, An n <> 0) ->
    Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
    Rabs x < / k -> { l:R | Pser An x l }.