# mathlib3documentation

analysis.p_series

# Convergence of p-series #

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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} {f : M} (hf : ⦃m n : ⦄, 0 < m m n f n f m) (n : ) :
(2 ^ n)).sum (λ (k : ), f k) (finset.range n).sum (λ (k : ), 2 ^ k f (2 ^ k))
theorem finset.le_sum_condensed {M : Type u_1} {f : M} (hf : ⦃m n : ⦄, 0 < m m n f n f m) (n : ) :
(finset.range (2 ^ n)).sum (λ (k : ), f k) f 0 + (finset.range n).sum (λ (k : ), 2 ^ k f (2 ^ k))
theorem finset.sum_condensed_le' {M : Type u_1} {f : M} (hf : ⦃m n : ⦄, 1 < m m n f n f m) (n : ) :
(finset.range n).sum (λ (k : ), 2 ^ k f (2 ^ (k + 1))) (2 ^ n + 1)).sum (λ (k : ), f k)
theorem finset.sum_condensed_le {M : Type u_1} {f : M} (hf : ⦃m n : ⦄, 1 < m m n f n f m) (n : ) :
(finset.range (n + 1)).sum (λ (k : ), 2 ^ k f (2 ^ k)) f 1 + 2 (2 ^ n + 1)).sum (λ (k : ), f k)
theorem ennreal.le_tsum_condensed {f : ennreal} (hf : ⦃m n : ⦄, 0 < m m n f 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 < m m n f n f m) :
∑' (k : ), 2 ^ k * f (2 ^ k) f 1 + 2 * ∑' (k : ), f k
theorem nnreal.summable_condensed_iff {f : nnreal} (hf : ⦃m n : ⦄, 0 < m m n f n f m) :
summable (λ (k : ), 2 ^ k * f (2 ^ k))

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 < m m n f n f m) :
summable (λ (k : ), 2 ^ k * f (2 ^ k))

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 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 : } :
summable (λ (n : ), n ^ p) p < -1
theorem real.summable_one_div_nat_rpow {p : } :
summable (λ (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.

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

Summability of the p-series over ℤ.

theorem real.summable_abs_int_rpow {b : } (hb : 1 < b) :
summable (λ (n : ), |n| ^ -b)

Harmonic series is not unconditionally summable.

Harmonic series is not unconditionally summable.

Divergence of the Harmonic Series

@[simp]
theorem nnreal.summable_rpow_inv {p : } :
summable (λ (n : ), (n ^ p)⁻¹) 1 < p
@[simp]
theorem nnreal.summable_rpow {p : } :
summable (λ (n : ), n ^ p) p < -1
theorem nnreal.summable_one_div_rpow {p : } :
summable (λ (n : ), 1 / n ^ p) 1 < p
theorem sum_Ioc_inv_sq_le_sub {α : Type u_1} {k n : } (hk : k 0) (h : k n) :
n).sum (λ (i : ), (i ^ 2)⁻¹) (k)⁻¹ - (n)⁻¹
theorem sum_Ioo_inv_sq_le {α : Type u_1} (k n : ) :
n).sum (λ (i : ), (i ^ 2)⁻¹) 2 / (k + 1)