Convergence of p-series

In this file we prove that the series ∑' k in ℕ, 1 / k ^ p converges if and only if p > 1. The proof is based on the Cauchy condensation test: ∑ k, f k converges if and only if so does ∑ k, 2 ^ k f (2 ^ k). We prove this test in NNReal.summable_condensed_iff and summable_condensed_iff_of_nonneg, then use it to prove summable_one_div_rpow. After this transformation, a p-series turns into a geometric series.

TODO

It should be easy to generalize arguments to Schlömilch's generalization of the Cauchy condensation test once we need it.

Tags

p-series, Cauchy condensation test

Cauchy condensation test

In this section we prove the Cauchy condensation test: for an antitone f : ℕ → ℝ≥0 or f : ℕ → ℝ, ∑ k, f k converges if and only if so does ∑ k, 2 ^ k f (2 ^ k). Instead of giving a monolithic proof, we split it into a series of lemmas with explicit estimates of partial sums of each series in terms of the partial sums of the other series.

theorem Finset.le_sum_condensed' {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) : (Finset.sum (Finset.Ico 1 (2 ^ n)) fun (k : ℕ) => f k) ≤ Finset.sum (Finset.range n) fun (k : ℕ) => 2 ^ k • f (2 ^ k)

theorem Finset.le_sum_condensed {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) : (Finset.sum (Finset.range (2 ^ n)) fun (k : ℕ) => f k) ≤ f 0 + Finset.sum (Finset.range n) fun (k : ℕ) => 2 ^ k • f (2 ^ k)

theorem Finset.sum_condensed_le' {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) : (Finset.sum (Finset.range n) fun (k : ℕ) => 2 ^ k • f (2 ^ (k + 1))) ≤ Finset.sum (Finset.Ico 2 (2 ^ n + 1)) fun (k : ℕ) => f k

theorem Finset.sum_condensed_le {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) : (Finset.sum (Finset.range (n + 1)) fun (k : ℕ) => 2 ^ k • f (2 ^ k)) ≤ f 1 + 2 • Finset.sum (Finset.Ico 2 (2 ^ n + 1)) fun (k : ℕ) => f k

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

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

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

Cauchy condensation test for a series of NNReal version.

theorem summable_condensed_iff_of_nonneg {f : ℕ → ℝ} (h_nonneg : ∀ (n : ℕ), 0 ≤ f n) (h_mono : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) : (Summable fun (k : ℕ) => 2 ^ k * f (2 ^ k)) ↔ Summable f

Cauchy condensation test for antitone series of nonnegative real numbers.

Convergence of the p-series

In this section we prove that for a real number p, the series ∑' n : ℕ, 1 / (n ^ p) converges if and only if 1 < p. There are many different proofs of this fact. The proof in this file uses the Cauchy condensation test we formalized above. This test implies that ∑ n, 1 / (n ^ p) converges if and only if ∑ n, 2 ^ n / ((2 ^ n) ^ p) converges, and the latter series is a geometric series with common ratio 2 ^ {1 - p}.

@[simp]

theorem Real.summable_nat_rpow_inv {p : ℝ} : (Summable fun (n : ℕ) => (↑n ^ p)⁻¹) ↔ 1 < p

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 Real.summable_nat_rpow {p : ℝ} : (Summable fun (n : ℕ) => ↑n ^ p) ↔ p < -1

theorem Real.summable_one_div_nat_rpow {p : ℝ} : (Summable fun (n : ℕ) => 1 / ↑n ^ p) ↔ 1 < p

Test for convergence of the p-series: the real-valued series ∑' n : ℕ, 1 / n ^ p converges if and only if 1 < p.

theorem Real.summable_nat_pow_inv {p : ℕ} : (Summable fun (n : ℕ) => (↑n ^ p)⁻¹) ↔ 1 < p

Test for convergence of the p-series: the real-valued series ∑' n : ℕ, (n ^ p)⁻¹ converges if and only if 1 < p.

theorem Real.summable_one_div_nat_pow {p : ℕ} : (Summable fun (n : ℕ) => 1 / ↑n ^ p) ↔ 1 < p

Test for convergence of the p-series: the real-valued series ∑' n : ℕ, 1 / n ^ p converges if and only if 1 < p.

theorem Real.summable_one_div_int_pow {p : ℕ} : (Summable fun (n : ℤ) => 1 / ↑n ^ p) ↔ 1 < p

Summability of the p-series over ℤ.

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

theorem Real.not_summable_nat_cast_inv : ¬Summable fun (n : ℕ) => (↑n)⁻¹

Harmonic series is not unconditionally summable.

theorem Real.not_summable_one_div_nat_cast : ¬Summable fun (n : ℕ) => 1 / ↑n

Harmonic series is not unconditionally summable.

theorem Real.tendsto_sum_range_one_div_nat_succ_atTop : Filter.Tendsto (fun (n : ℕ) => Finset.sum (Finset.range n) fun (i : ℕ) => 1 / (↑i + 1)) Filter.atTop Filter.atTop

Divergence of the Harmonic Series

@[simp]

theorem NNReal.summable_rpow_inv {p : ℝ} : (Summable fun (n : ℕ) => (↑n ^ p)⁻¹) ↔ 1 < p

@[simp]

theorem NNReal.summable_rpow {p : ℝ} : (Summable fun (n : ℕ) => ↑n ^ p) ↔ p < -1

theorem NNReal.summable_one_div_rpow {p : ℝ} : (Summable fun (n : ℕ) => 1 / ↑n ^ p) ↔ 1 < p

theorem sum_Ioc_inv_sq_le_sub {α : Type u_1} [LinearOrderedField α] {k : ℕ} {n : ℕ} (hk : k ≠ 0) (h : k ≤ n) : (Finset.sum (Finset.Ioc k n) fun (i : ℕ) => (↑i ^ 2)⁻¹) ≤ (↑k)⁻¹ - (↑n)⁻¹

theorem sum_Ioo_inv_sq_le {α : Type u_1} [LinearOrderedField α] (k : ℕ) (n : ℕ) : (Finset.sum (Finset.Ioo k n) fun (i : ℕ) => (↑i ^ 2)⁻¹) ≤ 2 / (↑k + 1)

theorem Real.not_summable_indicator_one_div_natCast {m : ℕ} (hm : m ≠ 0) (k : ZMod m) : ¬Summable (Set.indicator {n : ℕ | ↑n = k} fun (n : ℕ) => 1 / ↑n)

The harmonic series restricted to a residue class is not summable.

Translating the p-series by a real number

theorem Real.summable_one_div_nat_add_rpow (a : ℝ) (s : ℝ) : (Summable fun (n : ℕ) => 1 / |↑n + a| ^ s) ↔ 1 < s

theorem Real.summable_one_div_int_add_rpow (a : ℝ) (s : ℝ) : (Summable fun (n : ℤ) => 1 / |↑n + a| ^ s) ↔ 1 < s