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analysis.p_series

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 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} [ordered_add_comm_monoid M] {f : → M} (hf : ∀ ⦃m n : ⦄, 0 < mm nf n f m) (n : ) :
∑ (k : ) in finset.Ico 1 (2 ^ n), f k ∑ (k : ) in finset.range n, 2 ^ k •ℕ f (2 ^ k)

theorem finset.le_sum_condensed {M : Type u_1} [ordered_add_comm_monoid M] {f : → M} (hf : ∀ ⦃m n : ⦄, 0 < mm nf n f m) (n : ) :
∑ (k : ) in finset.range (2 ^ n), f k f 0 + ∑ (k : ) in finset.range n, 2 ^ k •ℕ f (2 ^ k)

theorem finset.sum_condensed_le' {M : Type u_1} [ordered_add_comm_monoid M] {f : → M} (hf : ∀ ⦃m n : ⦄, 1 < mm nf n f m) (n : ) :
∑ (k : ) in finset.range n, 2 ^ k •ℕ f (2 ^ (k + 1)) ∑ (k : ) in finset.Ico 2 (2 ^ n + 1), f k

theorem finset.sum_condensed_le {M : Type u_1} [ordered_add_comm_monoid M] {f : → M} (hf : ∀ ⦃m n : ⦄, 1 < mm nf n f m) (n : ) :
∑ (k : ) in finset.range (n + 1), 2 ^ k •ℕ f (2 ^ k) f 1 + 2 •ℕ ∑ (k : ) in finset.Ico 2 (2 ^ n + 1), f k

theorem ennreal.le_tsum_condensed {f : ennreal} (hf : ∀ ⦃m n : ⦄, 0 < mm nf n f m) :
(∑' (k : ), f k) f 0 + ∑' (k : ), (2 ^ k) * f (2 ^ k)

theorem ennreal.tsum_condensed_le {f : ennreal} (hf : ∀ ⦃m n : ⦄, 1 < mm nf n f m) :
(∑' (k : ), (2 ^ k) * f (2 ^ k)) f 1 + 2 * ∑' (k : ), f k

theorem nnreal.summable_condensed_iff {f : ℝ≥0} (hf : ∀ ⦃m n : ⦄, 0 < mm nf n f m) :
summable (λ (k : ), (2 ^ k) * f (2 ^ k)) summable f

Cauchy condensation test for a series of nnreal version.

theorem summable_condensed_iff_of_nonneg {f : } (h_nonneg : ∀ (n : ), 0 f n) (h_mono : ∀ ⦃m n : ⦄, 0 < mm nf n f m) :
summable (λ (k : ), (2 ^ k) * f (2 ^ k)) summable f

Cauchy condensation test for 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 : } :
summable (λ (n : ), (n ^ p)⁻¹) 1 < p

Test for congergence 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_rpow {p : } :
summable (λ (n : ), 1 / n ^ p) 1 < p

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

@[simp]
theorem real.summable_nat_pow_inv {p : } :
summable (λ (n : ), (n ^ p)⁻¹) 1 < p

Test for congergence 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 : } :
summable (λ (n : ), 1 / n ^ p) 1 < p

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

Harmonic series is not unconditionally summable.

Harmonic series is not unconditionally summable.

Harmonic series diverges.

@[simp]
theorem nnreal.summable_one_rpow_inv {p : } :
summable (λ (n : ), (n ^ p)⁻¹) 1 < p

theorem nnreal.summable_one_div_rpow {p : } :
summable (λ (n : ), 1 / n ^ p) 1 < p