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new series convergence criteria #2035

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nmolinamounier Jul 15, 2026
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new series convergence criteria
nmolinamounier Jul 15, 2026
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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -4,6 +4,9 @@

### Added

- in `sequences.v`,
+ lemmas `is_cvg_series_shiftn`, `series_near_le_cvg`

### Changed

### Renamed
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43 changes: 38 additions & 5 deletions theories/sequences.v
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Expand Up @@ -558,7 +558,7 @@ Proof. by rewrite telescopeK/= addrC addrNK. Qed.

Section series_patched.
Context (N : nat) {K : numFieldType} {V : normedModType K}.
Implicit Types (f : nat -> V) (u : V ^nat) (l : set_system V).
Implicit Types (f : nat -> V) (u : V ^nat) (l : set_system V).

Lemma is_cvg_series_restrict u_ :
cvgn [sequence \sum_(N <= k < n) u_ k]_n = cvgn (series u_).
Expand All @@ -571,6 +571,20 @@ rewrite funeqE => n; case: leqP => // ltNn; apply: (canRL (addrK _)).
by rewrite seriesEnat addrC -big_cat_nat// ltnW.
Qed.

Lemma is_cvg_series_shiftn u_ : cvgn (series u_) <-> cvgn [series u_ (n + N)]_n.
Proof.
split.
- rewrite -is_cvg_series_restrict => /cvg_ex[/= l +].
rewrite -(cvg_shiftn N)/= => Nnul; apply: cvgP; apply: cvg_trans Nnul.
apply: near_eq_cvg; near=> n.
rewrite /series/=.
by rewrite -{1}(add0n N) big_addn addnK.
- move=> cvgu; rewrite -is_cvg_series_restrict.
apply: cvgP; rewrite -(cvg_shiftn N)/=; apply: cvg_trans cvgu.
apply: near_eq_cvg; near=> n => /=.
by rewrite -{2}(add0n N) big_addn addnK.
Unshelve. all: by end_near. Qed.

End series_patched.

Section sequences_R_lemmas.
Expand Down Expand Up @@ -972,17 +986,36 @@ have := su_cv; rewrite near_swap => su_cvC; near=> m => /=; rewrite sub_series.
by have [|/ltnW]:= leqP m.2 m.1 => m12; rewrite ?normrN; near: m.
Unshelve. all: by end_near. Qed.

Lemma series_le_cvg (R : realType) (u_ v_ : R ^nat) :
(forall n, 0 <= u_ n) -> (forall n, 0 <= v_ n) ->
(forall n, u_ n <= v_ n) ->
Lemma series_le_cvg {R : realType} (u_ v_ : R ^nat) :

@affeldt-aist affeldt-aist Sep 27, 2026

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Rename to series_squeeze_is_cvgn?

(forall n, 0 <= u_ n) -> (forall n, 0 <= v_ n) ->
(forall n, u_ n <= v_ n) ->
cvgn (series v_) -> cvgn (series u_).
Proof.
move=> u_ge0 v_ge0 le_uv /cvg_seq_bounded/bounded_fun_has_ubound[M v_M].
apply: nondecreasing_is_cvgn; first exact: nondecreasing_series.
exists M => _ [n _ <-].
by apply: le_trans (v_M (series v_ n) _); [apply: ler_sum | exists n].
by apply: le_trans (v_M (series v_ n) _); [exact: ler_sum | exists n].
Qed.

Lemma series_near_le_cvg {R : realType} (u_ v_ : R^nat) :

@affeldt-aist affeldt-aist Sep 27, 2026 •
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Maybe near_series_squeeze_is_cvgn is better because it follows the pattern of the squeeze lemmas (and anyway it should contain the is_cvg substring).

(\forall n \near \oo, 0 <= u_ n) -> (\forall n \near \oo, 0 <= v_ n) ->
(\forall n \near \oo, u_ n <= v_ n) ->
cvgn (series v_) -> cvgn (series u_).
Proof.
move=> u0 v0 uv cvg_v.
near \oo => N; apply/(is_cvg_series_shiftn N).
move: cvg_v => /(is_cvg_series_shiftn N); apply: series_le_cvg => /= n.
- have : forall n, (n >= N)%N -> 0 <= u_ n.
by near: N; exact: (iffLR (near_infty_leq _)).
by apply; exact: leq_addl.
- have : forall n, (n >= N)%N -> 0 <= v_ n.
by near: N; exact: (iffLR (near_infty_leq _)).
by apply; exact: leq_addl.
- have : forall n, (n >= N)%N -> u_ n <= v_ n.
by near: N; exact: (iffLR (near_infty_leq _)).
by apply; exact: leq_addl.
Unshelve. all: by end_near. Qed.

Lemma normed_cvg {R : realType} (V : completeNormedModType R) (u_ : V ^nat) :
cvgn [normed series u_] -> cvgn (series u_).
Proof.
Expand Down
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