File: Mathlib/Analysis/PSeries.lean

full_name: SuccDiffBounded
code: /--
A sequence `u` has the property that its ratio of successive differences is bounded
when there is a positive real number `C` such that, for all n ∈ ℕ,
(u (n + 2) - u (n + 1)) ≤ C * (u (n + 1) - u n)
-/
def SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop 
goal: Prop

full_name: Finset.le_sum_schlomilch'
code: theorem le_sum_schlomilch' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ Ico (u 0) (u n), f k) ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) 
goal: ∑ k ∈ Finset.Ico (u 0) (u n), f k ≤ ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u k)

full_name: Finset.le_sum_condensed'
code: theorem le_sum_condensed' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ Ico 1 (2 ^ n), f k) ≤ ∑ k ∈ range n, 2 ^ k • f (2 ^ k) 
goal: ∑ k ∈ Finset.Ico 1 (2 ^ n), f k ≤ ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ k)

full_name: Finset.le_sum_schlomilch
code: theorem le_sum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ range (u n), f k) ≤
      ∑ k ∈ range (u 0), f k + ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) 
goal: ∑ k ∈ Finset.range (u n), f k ≤ ∑ k ∈ Finset.range (u 0), f k + ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u k)

full_name: Finset.le_sum_condensed
code: theorem le_sum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ range (2 ^ n), f k) ≤ f 0 + ∑ k ∈ range n, 2 ^ k • f (2 ^ k) 
goal: ∑ k ∈ Finset.range (2 ^ n), f k ≤ f 0 + ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ k)

full_name: Finset.sum_schlomilch_le'
code: theorem sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k 
goal: ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Finset.Ico (u 0 + 1) (u n + 1), f k

full_name: Finset.sum_condensed_le'
code: theorem sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))) ≤ ∑ k ∈ Ico 2 (2 ^ n + 1), f k 
goal: ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ (k + 1)) ≤ ∑ k ∈ Finset.Ico 2 (2 ^ n + 1), f k

full_name: Finset.sum_schlomilch_le
code: theorem sum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) (n : ℕ) :
    ∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u k) ≤
    (u 1 - u 0) • f (u 0) + C • ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k 
goal: ∑ k ∈ Finset.range (n + 1), (u (k + 1) - u k) • f (u k) ≤
  (u 1 - u 0) • f (u 0) + C • ∑ k ∈ Finset.Ico (u 0 + 1) (u n + 1), f k

full_name: Finset.sum_condensed_le
code: theorem sum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k)) ≤ f 1 + 2 • ∑ k ∈ Ico 2 (2 ^ n + 1), f k 
goal: ∑ k ∈ Finset.range (n + 1), 2 ^ k • f (2 ^ k) ≤ f 1 + 2 • ∑ k ∈ Finset.Ico 2 (2 ^ n + 1), f k

full_name: ENNReal.le_tsum_schlomilch
code: theorem le_tsum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : StrictMono u) :
    ∑' k , f k ≤ ∑ k ∈ range (u 0), f k + ∑' k : ℕ, (u (k + 1) - u k) * f (u k) 
goal: ∑' (k : ℕ), f k ≤ ∑ k ∈ Finset.range (u 0), f k + ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k)

full_name: ENNReal.le_tsum_condensed
code: theorem le_tsum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
    ∑' k, f k ≤ f 0 + ∑' k : ℕ, 2 ^ k * f (2 ^ k) 
goal: ∑' (k : ℕ), f k ≤ f 0 + ∑' (k : ℕ), 2 ^ k * f (2 ^ k)

full_name: ENNReal.tsum_schlomilch_le
code: theorem tsum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) :
    ∑' k : ℕ, (u (k + 1) - u k) * f (u k) ≤ (u 1 - u 0) * f (u 0) + C * ∑' k, f k 
goal: ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k

full_name: ENNReal.tsum_condensed_le
code: theorem tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) :
    (∑' k : ℕ, 2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' k, f k 
goal: ∑' (k : ℕ), 2 ^ k * f (2 ^ k) ≤ f 1 + 2 * ∑' (k : ℕ), f k

full_name: NNReal.summable_schlomilch_iff
code: /-- for a series of `NNReal` version. -/
theorem summable_schlomilch_iff {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ≥0}
    (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m)
    (h_pos : ∀ n, 0 < u n) (hu_strict : StrictMono u)
    (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
    (Summable fun k : ℕ => (u (k + 1) - (u k : ℝ≥0)) * f (u k)) ↔ Summable f 
goal: (Summable fun k => (↑(u (k + 1)) - ↑(u k)) * f (u k)) ↔ Summable f

full_name: NNReal.summable_condensed_iff
code: theorem summable_condensed_iff {f : ℕ → ℝ≥0} (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
    (Summable fun k : ℕ => (2 : ℝ≥0) ^ k * f (2 ^ k)) ↔ Summable f 
goal: (Summable fun k => 2 ^ k * f (2 ^ k)) ↔ Summable f

full_name: summable_schlomilch_iff_of_nonneg
code: /-- for series of nonnegative real numbers. -/
theorem summable_schlomilch_iff_of_nonneg {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
    (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
    (Summable fun k : ℕ => (u (k + 1) - (u k : ℝ)) * f (u k)) ↔ Summable f 
goal: (Summable fun k => (↑(u (k + 1)) - ↑(u k)) * f (u k)) ↔ Summable f

full_name: summable_condensed_iff_of_nonneg
code: /-- Cauchy condensation test for antitone series of nonnegative real numbers. -/
theorem summable_condensed_iff_of_nonneg {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
    (h_mono : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
    (Summable fun k : ℕ => (2 : ℝ) ^ k * f (2 ^ k)) ↔ Summable f 
goal: (Summable fun k => 2 ^ k * f (2 ^ k)) ↔ Summable f

full_name: Real.summable_nat_rpow_inv
code: /-- Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, (n ^ p)⁻¹` converges
if and only if `1 < p`. -/
@[simp]
theorem summable_nat_rpow_inv {p : ℝ} :
    Summable (fun n => ((n : ℝ) ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p 
goal: (Summable fun n => (↑n ^ p)⁻¹) ↔ 1 < p

full_name: Real.summable_nat_rpow
code: @[simp]
theorem summable_nat_rpow {p : ℝ} : Summable (fun n => (n : ℝ) ^ p : ℕ → ℝ) ↔ p < -1 
goal: (Summable fun n => ↑n ^ p) ↔ p < -1

full_name: Real.summable_one_div_nat_rpow
code: /-- Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, 1 / n ^ p` converges
if and only if `1 < p`. -/
theorem summable_one_div_nat_rpow {p : ℝ} :
    Summable (fun n => 1 / (n : ℝ) ^ p : ℕ → ℝ) ↔ 1 < p 
goal: (Summable fun n => 1 / ↑n ^ p) ↔ 1 < p

full_name: Real.summable_nat_pow_inv
code: /-- Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, (n ^ p)⁻¹` converges
if and only if `1 < p`. -/
@[simp]
theorem summable_nat_pow_inv {p : ℕ} :
    Summable (fun n => ((n : ℝ) ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p 
goal: (Summable fun n => (↑n ^ p)⁻¹) ↔ 1 < p

full_name: Real.summable_one_div_nat_pow
code: /-- Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, 1 / n ^ p` converges
if and only if `1 < p`. -/
theorem summable_one_div_nat_pow {p : ℕ} :
    Summable (fun n => 1 / (n : ℝ) ^ p : ℕ → ℝ) ↔ 1 < p 
goal: (Summable fun n => 1 / ↑n ^ p) ↔ 1 < p

full_name: Real.summable_one_div_int_pow
code: /-- Summability of the `p`-series over `ℤ`. -/
theorem summable_one_div_int_pow {p : ℕ} :
    (Summable fun n : ℤ ↦ 1 / (n : ℝ) ^ p) ↔ 1 < p 
goal: (Summable fun n => 1 / ↑n ^ p) ↔ 1 < p

full_name: Real.summable_abs_int_rpow
code: theorem summable_abs_int_rpow {b : ℝ} (hb : 1 < b) :
    Summable fun n : ℤ => |(n : ℝ)| ^ (-b) 
goal: Summable fun n => |↑n| ^ (-b)

full_name: Real.not_summable_natCast_inv
code: /-- Harmonic series is not unconditionally summable. -/
theorem not_summable_natCast_inv : ¬Summable (fun n => n⁻¹ : ℕ → ℝ) 
goal: ¬Summable fun n => (↑n)⁻¹

full_name: Real.not_summable_one_div_natCast
code: /-- Harmonic series is not unconditionally summable. -/
theorem not_summable_one_div_natCast : ¬Summable (fun n => 1 / n : ℕ → ℝ) 
goal: ¬Summable fun n => 1 / ↑n

full_name: Real.tendsto_sum_range_one_div_nat_succ_atTop
code: /-- **Divergence of the Harmonic Series** -/
theorem tendsto_sum_range_one_div_nat_succ_atTop :
    Tendsto (fun n => ∑ i ∈ Finset.range n, (1 / (i + 1) : ℝ)) atTop atTop 
goal: Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, 1 / (↑i + 1)) Filter.atTop Filter.atTop

full_name: NNReal.summable_rpow_inv
code: @[simp]
theorem summable_rpow_inv {p : ℝ} :
    Summable (fun n => ((n : ℝ≥0) ^ p)⁻¹ : ℕ → ℝ≥0) ↔ 1 < p 
goal: (Summable fun n => (↑n ^ p)⁻¹) ↔ 1 < p

full_name: NNReal.summable_rpow
code: @[simp]
theorem summable_rpow {p : ℝ} : Summable (fun n => (n : ℝ≥0) ^ p : ℕ → ℝ≥0) ↔ p < -1 
goal: (Summable fun n => ↑n ^ p) ↔ p < -1

full_name: NNReal.summable_one_div_rpow
code: theorem summable_one_div_rpow {p : ℝ} :
    Summable (fun n => 1 / (n : ℝ≥0) ^ p : ℕ → ℝ≥0) ↔ 1 < p 
goal: (Summable fun n => 1 / ↑n ^ p) ↔ 1 < p

full_name: sum_Ioc_inv_sq_le_sub
code: theorem sum_Ioc_inv_sq_le_sub {k n : ℕ} (hk : k ≠ 0) (h : k ≤ n) :
    (∑ i ∈ Ioc k n, ((i : α) ^ 2)⁻¹) ≤ (k : α)⁻¹ - (n : α)⁻¹ 
goal: ∑ i ∈ Finset.Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹

full_name: sum_Ioo_inv_sq_le
code: theorem sum_Ioo_inv_sq_le (k n : ℕ) : (∑ i ∈ Ioo k n, (i ^ 2 : α)⁻¹) ≤ 2 / (k + 1) 
goal: ∑ i ∈ Finset.Ioo k n, (↑i ^ 2)⁻¹ ≤ 2 / (↑k + 1)

full_name: Real.not_summable_indicator_one_div_natCast
code: /-- The harmonic series restricted to a residue class is not summable. -/
lemma Real.not_summable_indicator_one_div_natCast {m : ℕ} (hm : m ≠ 0) (k : ZMod m) :
    ¬ Summable ({n : ℕ | (n : ZMod m) = k}.indicator fun n : ℕ ↦ (1 / n : ℝ)) 
goal: ¬Summable ({n | ↑n = k}.indicator fun n => 1 / ↑n)

full_name: Real.summable_one_div_nat_add_rpow
code: lemma Real.summable_one_div_nat_add_rpow (a : ℝ) (s : ℝ) :
    Summable (fun n : ℕ ↦ 1 / |n + a| ^ s) ↔ 1 < s 
goal: (Summable fun n => 1 / |↑n + a| ^ s) ↔ 1 < s

full_name: Real.summable_one_div_int_add_rpow
code: lemma Real.summable_one_div_int_add_rpow (a : ℝ) (s : ℝ) :
    Summable (fun n : ℤ ↦ 1 / |n + a| ^ s) ↔ 1 < s 
goal: (Summable fun n => 1 / |↑n + a| ^ s) ↔ 1 < s