A collection of specific limit computations

This file contains important specific limit computations in (semi-)normed groups/rings/spaces, as well as such computations in ℝ when the natural proof passes through a fact about normed spaces.

theorem tendsto_norm_atTop_atTop : Filter.Tendsto norm Filter.atTop Filter.atTop

theorem summable_of_absolute_convergence_real {f : ℕ → ℝ} : (∃ r, Filter.Tendsto (fun n => Finset.sum (Finset.range n) fun i => |f i|) Filter.atTop (nhds r)) → Summable f

Powers

theorem tendsto_norm_zero' {𝕜 : Type u_4} [NormedAddCommGroup 𝕜] : Filter.Tendsto norm (nhdsWithin 0 {0}ᶜ) (nhdsWithin 0 (Set.Ioi 0))

theorem NormedField.tendsto_norm_inverse_nhdsWithin_0_atTop {𝕜 : Type u_4} [NormedField 𝕜] : Filter.Tendsto (fun x => ‖x⁻¹‖) (nhdsWithin 0 {0}ᶜ) Filter.atTop

theorem NormedField.tendsto_norm_zpow_nhdsWithin_0_atTop {𝕜 : Type u_4} [NormedField 𝕜] {m : ℤ} (hm : m < 0) : Filter.Tendsto (fun x => ‖x ^ m‖) (nhdsWithin 0 {0}ᶜ) Filter.atTop

theorem NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded {ι : Type u_4} {𝕜 : Type u_5} {𝔸 : Type u_6} [NormedField 𝕜] [NormedAddCommGroup 𝔸] [NormedSpace 𝕜 𝔸] {l : Filter ι} {ε : ι → 𝕜} {f : ι → 𝔸} (hε : Filter.Tendsto ε l (nhds 0)) (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)) : Filter.Tendsto (ε • f) l (nhds 0)

The (scalar) product of a sequence that tends to zero with a bounded one also tends to zero.

@[simp]

theorem NormedField.continuousAt_zpow {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {m : ℤ} {x : 𝕜} : ContinuousAt (fun x => x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m

@[simp]

theorem NormedField.continuousAt_inv {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {x : 𝕜} : ContinuousAt Inv.inv x ↔ x ≠ 0

theorem isLittleO_pow_pow_of_lt_left {r₁ : ℝ} {r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ < r₂) : (fun n => r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n

theorem isBigO_pow_pow_of_le_left {r₁ : ℝ} {r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ ≤ r₂) : (fun n => r₁ ^ n) =O[Filter.atTop] fun n => r₂ ^ n

theorem isLittleO_pow_pow_of_abs_lt_left {r₁ : ℝ} {r₂ : ℝ} (h : |r₁| < |r₂|) : (fun n => r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n

theorem TFAE_exists_lt_isLittleO_pow (f : ℕ → ℝ) (R : ℝ) : List.TFAE [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[Filter.atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[Filter.atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[Filter.atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[Filter.atTop] fun x => a ^ x, ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n, ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n]

Various statements equivalent to the fact that f n grows exponentially slower than R ^ n.

• 0: $f n = o(a ^ n)$ for some $-R < a < R$;
• 1: $f n = o(a ^ n)$ for some $0 < a < R$;
• 2: $f n = O(a ^ n)$ for some $-R < a < R$;
• 3: $f n = O(a ^ n)$ for some $0 < a < R$;
• 4: there exist a < R and C such that one of C and R is positive and $|f n| ≤ Ca^n$ for all n;
• 5: there exists 0 < a < R and a positive C such that $|f n| ≤ Ca^n$ for all n;
• 6: there exists a < R such that $|f n| ≤ a ^ n$ for sufficiently large n;
• 7: there exists 0 < a < R such that $|f n| ≤ a ^ n$ for sufficiently large n.

NB: For backwards compatibility, if you add more items to the list, please append them at the end of the list.

theorem isLittleO_pow_const_const_pow_of_one_lt {R : Type u_4} [NormedRing R] (k : ℕ) {r : ℝ} (hr : 1 < r) : (fun n => ↑n ^ k) =o[Filter.atTop] fun n => r ^ n

For any natural k and a real r > 1 we have n ^ k = o(r ^ n) as n → ∞.

theorem isLittleO_coe_const_pow_of_one_lt {R : Type u_4} [NormedRing R] {r : ℝ} (hr : 1 < r) : Nat.cast =o[Filter.atTop] fun n => r ^ n

For a real r > 1 we have n = o(r ^ n) as n → ∞.

theorem isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt {R : Type u_4} [NormedRing R] (k : ℕ) {r₁ : R} {r₂ : ℝ} (h : ‖r₁‖ < r₂) : (fun n => ↑n ^ k * r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n

If ‖r₁‖ < r₂, then for any natural k we have n ^ k r₁ ^ n = o (r₂ ^ n) as n → ∞.

theorem tendsto_pow_const_div_const_pow_of_one_lt (k : ℕ) {r : ℝ} (hr : 1 < r) : Filter.Tendsto (fun n => ↑n ^ k / r ^ n) Filter.atTop (nhds 0)

theorem tendsto_pow_const_mul_const_pow_of_abs_lt_one (k : ℕ) {r : ℝ} (hr : |r| < 1) : Filter.Tendsto (fun n => ↑n ^ k * r ^ n) Filter.atTop (nhds 0)

If |r| < 1, then n ^ k r ^ n tends to zero for any natural k.

theorem tendsto_pow_const_mul_const_pow_of_lt_one (k : ℕ) {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1) : Filter.Tendsto (fun n => ↑n ^ k * r ^ n) Filter.atTop (nhds 0)

If 0 ≤ r < 1, then n ^ k r ^ n tends to zero for any natural k. This is a specialized version of tendsto_pow_const_mul_const_pow_of_abs_lt_one, singled out for ease of application.

theorem tendsto_self_mul_const_pow_of_abs_lt_one {r : ℝ} (hr : |r| < 1) : Filter.Tendsto (fun n => ↑n * r ^ n) Filter.atTop (nhds 0)

If |r| < 1, then n * r ^ n tends to zero.

theorem tendsto_self_mul_const_pow_of_lt_one {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1) : Filter.Tendsto (fun n => ↑n * r ^ n) Filter.atTop (nhds 0)

If 0 ≤ r < 1, then n * r ^ n tends to zero. This is a specialized version of tendsto_self_mul_const_pow_of_abs_lt_one, singled out for ease of application.

theorem tendsto_pow_atTop_nhds_0_of_norm_lt_1 {R : Type u_4} [NormedRing R] {x : R} (h : ‖x‖ < 1) : Filter.Tendsto (fun n => x ^ n) Filter.atTop (nhds 0)

In a normed ring, the powers of an element x with ‖x‖ < 1 tend to zero.

theorem tendsto_pow_atTop_nhds_0_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : Filter.Tendsto (fun n => r ^ n) Filter.atTop (nhds 0)

Geometric series

theorem hasSum_geometric_of_norm_lt_1 {K : Type u_4} [NormedField K] {ξ : K} (h : ‖ξ‖ < 1) : HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹

theorem summable_geometric_of_norm_lt_1 {K : Type u_4} [NormedField K] {ξ : K} (h : ‖ξ‖ < 1) : Summable fun n => ξ ^ n

theorem tsum_geometric_of_norm_lt_1 {K : Type u_4} [NormedField K] {ξ : K} (h : ‖ξ‖ < 1) : ∑' (n : ℕ), ξ ^ n = (1 - ξ)⁻¹

theorem hasSum_geometric_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : HasSum (fun n => r ^ n) (1 - r)⁻¹

theorem summable_geometric_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : Summable fun n => r ^ n

theorem tsum_geometric_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : ∑' (n : ℕ), r ^ n = (1 - r)⁻¹

@[simp]

theorem summable_geometric_iff_norm_lt_1 {K : Type u_4} [NormedField K] {ξ : K} : (Summable fun n => ξ ^ n) ↔ ‖ξ‖ < 1

A geometric series in a normed field is summable iff the norm of the common ratio is less than one.

theorem summable_norm_pow_mul_geometric_of_norm_lt_1 {R : Type u_4} [NormedRing R] (k : ℕ) {r : R} (hr : ‖r‖ < 1) : Summable fun n => ‖↑n ^ k * r ^ n‖

theorem summable_pow_mul_geometric_of_norm_lt_1 {R : Type u_4} [NormedRing R] [CompleteSpace R] (k : ℕ) {r : R} (hr : ‖r‖ < 1) : Summable fun n => ↑n ^ k * r ^ n

theorem hasSum_coe_mul_geometric_of_norm_lt_1 {𝕜 : Type u_4} [NormedField 𝕜] [CompleteSpace 𝕜] {r : 𝕜} (hr : ‖r‖ < 1) : HasSum (fun n => ↑n * r ^ n) (r / (1 - r) ^ 2)

If ‖r‖ < 1, then ∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2, HasSum version.

theorem tsum_coe_mul_geometric_of_norm_lt_1 {𝕜 : Type u_4} [NormedField 𝕜] [CompleteSpace 𝕜] {r : 𝕜} (hr : ‖r‖ < 1) : ∑' (n : ℕ), ↑n * r ^ n = r / (1 - r) ^ 2

If ‖r‖ < 1, then ∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2.

theorem SeminormedAddCommGroup.cauchySeq_of_le_geometric {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {r : ℝ} (hr : r < 1) {u : ℕ → α} (h : ∀ (n : ℕ), ‖u n - u (n + 1)‖ ≤ C * r ^ n) : CauchySeq u

theorem dist_partial_sum_le_of_le_geometric {α : Type u_1} [SeminormedAddCommGroup α] {r : ℝ} {C : ℝ} {f : ℕ → α} (hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n) (n : ℕ) : dist (Finset.sum (Finset.range n) fun i => f i) (Finset.sum (Finset.range (n + 1)) fun i => f i) ≤ C * r ^ n

theorem cauchySeq_finset_of_geometric_bound {α : Type u_1} [SeminormedAddCommGroup α] {r : ℝ} {C : ℝ} {f : ℕ → α} (hr : r < 1) (hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n) : CauchySeq fun s => Finset.sum s fun x => f x

If ‖f n‖ ≤ C * r ^ n for all n : ℕ and some r < 1, then the partial sums of f form a Cauchy sequence. This lemma does not assume 0 ≤ r or 0 ≤ C.

theorem norm_sub_le_of_geometric_bound_of_hasSum {α : Type u_1} [SeminormedAddCommGroup α] {r : ℝ} {C : ℝ} {f : ℕ → α} (hr : r < 1) (hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n) {a : α} (ha : HasSum f a) (n : ℕ) : ‖(Finset.sum (Finset.range n) fun x => f x) - a‖ ≤ C * r ^ n / (1 - r)

If ‖f n‖ ≤ C * r ^ n for all n : ℕ and some r < 1, then the partial sums of f are within distance C * r ^ n / (1 - r) of the sum of the series. This lemma does not assume 0 ≤ r or 0 ≤ C.

@[simp]

theorem dist_partial_sum {α : Type u_1} [SeminormedAddCommGroup α] (u : ℕ → α) (n : ℕ) : dist (Finset.sum (Finset.range (n + 1)) fun k => u k) (Finset.sum (Finset.range n) fun k => u k) = ‖u n‖

@[simp]

theorem dist_partial_sum' {α : Type u_1} [SeminormedAddCommGroup α] (u : ℕ → α) (n : ℕ) : dist (Finset.sum (Finset.range n) fun k => u k) (Finset.sum (Finset.range (n + 1)) fun k => u k) = ‖u n‖

theorem cauchy_series_of_le_geometric {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1) (h : ∀ (n : ℕ), ‖u n‖ ≤ C * r ^ n) : CauchySeq fun n => Finset.sum (Finset.range n) fun k => u k

theorem NormedAddCommGroup.cauchy_series_of_le_geometric' {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1) (h : ∀ (n : ℕ), ‖u n‖ ≤ C * r ^ n) : CauchySeq fun n => Finset.sum (Finset.range (n + 1)) fun k => u k

theorem NormedAddCommGroup.cauchy_series_of_le_geometric'' {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {u : ℕ → α} {N : ℕ} {r : ℝ} (hr₀ : 0 < r) (hr₁ : r < 1) (h : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n) : CauchySeq fun n => Finset.sum (Finset.range (n + 1)) fun k => u k

theorem NormedRing.summable_geometric_of_norm_lt_1 {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1) : Summable fun n => x ^ n

A geometric series in a complete normed ring is summable. Proved above (same name, different namespace) for not-necessarily-complete normed fields.

theorem NormedRing.tsum_geometric_of_norm_lt_1 {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1) : ‖∑' (n : ℕ), x ^ n‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹

Bound for the sum of a geometric series in a normed ring. This formula does not assume that the normed ring satisfies the axiom ‖1‖ = 1.

theorem geom_series_mul_neg {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1) : (∑' (i : ℕ), x ^ i) * (1 - x) = 1

theorem mul_neg_geom_series {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1) : (1 - x) * ∑' (i : ℕ), x ^ i = 1

Summability tests based on comparison with geometric series

theorem summable_of_ratio_norm_eventually_le {α : Type u_4} [SeminormedAddCommGroup α] [CompleteSpace α] {f : ℕ → α} {r : ℝ} (hr₁ : r < 1) (h : ∀ᶠ (n : ℕ) in Filter.atTop, ‖f (n + 1)‖ ≤ r * ‖f n‖) : Summable f

theorem summable_of_ratio_test_tendsto_lt_one {α : Type u_4} [NormedAddCommGroup α] [CompleteSpace α] {f : ℕ → α} {l : ℝ} (hl₁ : l < 1) (hf : ∀ᶠ (n : ℕ) in Filter.atTop, f n ≠ 0) (h : Filter.Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) Filter.atTop (nhds l)) : Summable f

theorem not_summable_of_ratio_norm_eventually_ge {α : Type u_4} [SeminormedAddCommGroup α] {f : ℕ → α} {r : ℝ} (hr : 1 < r) (hf : ∃ᶠ (n : ℕ) in Filter.atTop, ‖f n‖ ≠ 0) (h : ∀ᶠ (n : ℕ) in Filter.atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖) : ¬Summable f

theorem not_summable_of_ratio_test_tendsto_gt_one {α : Type u_4} [SeminormedAddCommGroup α] {f : ℕ → α} {l : ℝ} (hl : 1 < l) (h : Filter.Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) Filter.atTop (nhds l)) : ¬Summable f

Dirichlet and alternating series tests

theorem Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ℝ} {f : ℕ → ℝ} {z : ℕ → E} (hfa : Monotone f) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) (hgb : ∀ (n : ℕ), ‖Finset.sum (Finset.range n) fun i => z i‖ ≤ b) : CauchySeq fun n => Finset.sum (Finset.range (n + 1)) fun i => f i • z i

Dirichlet's Test for monotone sequences.

theorem Antitone.cauchySeq_series_mul_of_tendsto_zero_of_bounded {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ℝ} {f : ℕ → ℝ} {z : ℕ → E} (hfa : Antitone f) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) (hzb : ∀ (n : ℕ), ‖Finset.sum (Finset.range n) fun i => z i‖ ≤ b) : CauchySeq fun n => Finset.sum (Finset.range (n + 1)) fun i => f i • z i

Dirichlet's test for antitone sequences.

theorem norm_sum_neg_one_pow_le (n : ℕ) : ‖Finset.sum (Finset.range n) fun i => (-1) ^ i‖ ≤ 1

theorem Monotone.cauchySeq_alternating_series_of_tendsto_zero {f : ℕ → ℝ} (hfa : Monotone f) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) : CauchySeq fun n => Finset.sum (Finset.range (n + 1)) fun i => (-1) ^ i * f i

The alternating series test for monotone sequences. See also Monotone.tendsto_alternating_series_of_tendsto_zero.

theorem Monotone.tendsto_alternating_series_of_tendsto_zero {f : ℕ → ℝ} (hfa : Monotone f) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) : ∃ l, Filter.Tendsto (fun n => Finset.sum (Finset.range (n + 1)) fun i => (-1) ^ i * f i) Filter.atTop (nhds l)

The alternating series test for monotone sequences.

theorem Antitone.cauchySeq_alternating_series_of_tendsto_zero {f : ℕ → ℝ} (hfa : Antitone f) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) : CauchySeq fun n => Finset.sum (Finset.range (n + 1)) fun i => (-1) ^ i * f i

The alternating series test for antitone sequences. See also Antitone.tendsto_alternating_series_of_tendsto_zero.

theorem Antitone.tendsto_alternating_series_of_tendsto_zero {f : ℕ → ℝ} (hfa : Antitone f) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) : ∃ l, Filter.Tendsto (fun n => Finset.sum (Finset.range (n + 1)) fun i => (-1) ^ i * f i) Filter.atTop (nhds l)

The alternating series test for antitone sequences.

Factorial

theorem Real.summable_pow_div_factorial (x : ℝ) : Summable fun n => x ^ n / ↑(Nat.factorial n)

The series ∑' n, x ^ n / n! is summable of any x : ℝ. See also exp_series_div_summable for a version that also works in ℂ, and exp_series_summable' for a version that works in any normed algebra over ℝ or ℂ.

theorem Real.tendsto_pow_div_factorial_atTop (x : ℝ) : Filter.Tendsto (fun n => x ^ n / ↑(Nat.factorial n)) Filter.atTop (nhds 0)