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feat(Analysis/PSeries): add version of Cauchy condensation with weaker assumptions #34985

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feat(Analysis/PSeries): add version of Cauchy condensation with weaker assumptions #34985

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16 changes: 16 additions & 0 deletions Mathlib/Analysis/PSeries.lean
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Expand Up @@ -235,6 +235,22 @@ theorem summable_condensed_iff_of_nonneg {f : ℕ → ℝ} (h_nonneg : ∀ n, 0
(pow_right_strictMono₀ one_lt_two) two_ne_zero h_succ_diff
simp [pow_succ, mul_two]

/-- Cauchy condensation test for eventually antitone and nonnegative series of real numbers. -/
theorem summable_condensed_iff_of_nonneg' {f : ℕ → ℝ} (h_nonneg : 0 ≤ᶠ[Filter.atTop] f)
(h_mono : ∃ k : ℕ, AntitoneOn f (Set.Ici k)) :
(Summable fun k : ℕ => (2 : ℝ) ^ k * f (2 ^ k)) ↔ Summable f := by
rw [Filter.EventuallyLE, Filter.eventually_atTop] at h_nonneg
rcases h_nonneg with ⟨n,hn⟩
rcases h_mono with ⟨m,hm⟩
convert summable_condensed_iff_of_nonneg (f := fun k ↦ f (max k (n + m))) _ _ using 1
· rw [summable_congr_atTop]
have h_pow := tendsto_pow_atTop_atTop_of_one_lt (r := 2) (by simp)
filter_upwards [h_pow.eventually_ge_atTop (n + m)] with _ hk using by simp [max_eq_left hk]
· rw [summable_congr_atTop]
filter_upwards [Filter.eventually_ge_atTop (n + m)] with _ hk using by simp [max_eq_left hk]
· exact fun _ ↦ by apply hn; simp
· exact fun _ _ _ _ ↦ hm (by simp) (by simp) (by bound)

section p_series

/-!
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