# Documentation

Mathlib.Analysis.SpecificLimits.Normed

# 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 : } :
(r, Filter.Tendsto (fun n => Finset.sum () fun i => |f i|) Filter.atTop (nhds r)) →

### Powers #

theorem tendsto_norm_zero' {𝕜 : Type u_4} :
theorem NormedField.tendsto_norm_zpow_nhdsWithin_0_atTop {𝕜 : Type u_4} [] {m : } (hm : m < 0) :
Filter.Tendsto (fun x => x ^ m) (nhdsWithin 0 {0}) Filter.atTop
theorem NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded {ι : Type u_4} {𝕜 : Type u_5} {𝔸 : Type u_6} [] [] {l : } {ε : ι𝕜} {f : ι𝔸} (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} {m : } {x : 𝕜} :
ContinuousAt (fun x => x ^ m) x x 0 0 m
@[simp]
theorem NormedField.continuousAt_inv {𝕜 : Type u_4} {x : 𝕜} :
ContinuousAt Inv.inv x x 0
theorem isLittleO_pow_pow_of_lt_left {r₁ : } {r₂ : } (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 : } {r₂ : } (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 : } {r₂ : } (h : |r₁| < |r₂|) :
(fun n => r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n
theorem TFAE_exists_lt_isLittleO_pow (f : ) (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 : ), |f n| C * a ^ n, a, a Set.Ioo 0 R C, C > 0 ∀ (n : ), |f n| C * a ^ n, a, a < R ∀ᶠ (n : ) in Filter.atTop, |f n| a ^ n, a, a Set.Ioo 0 R ∀ᶠ (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} [] (k : ) {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} [] {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} [] (k : ) {r₁ : R} {r₂ : } (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 : ) {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 : ) {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 : ) {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 : } (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 : } (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} [] {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 : } (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} [] {ξ : K} (h : ξ < 1) :
HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹
theorem summable_geometric_of_norm_lt_1 {K : Type u_4} [] {ξ : K} (h : ξ < 1) :
Summable fun n => ξ ^ n
theorem tsum_geometric_of_norm_lt_1 {K : Type u_4} [] {ξ : K} (h : ξ < 1) :
∑' (n : ), ξ ^ n = (1 - ξ)⁻¹
theorem hasSum_geometric_of_abs_lt_1 {r : } (h : |r| < 1) :
HasSum (fun n => r ^ n) (1 - r)⁻¹
theorem summable_geometric_of_abs_lt_1 {r : } (h : |r| < 1) :
Summable fun n => r ^ n
theorem tsum_geometric_of_abs_lt_1 {r : } (h : |r| < 1) :
∑' (n : ), r ^ n = (1 - r)⁻¹
@[simp]
theorem summable_geometric_iff_norm_lt_1 {K : Type u_4} [] {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} [] (k : ) {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} [] [] (k : ) {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} [] [] {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} [] [] {r : 𝕜} (hr : r < 1) :
∑' (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} {C : } {r : } (hr : r < 1) {u : α} (h : ∀ (n : ), u n - u (n + 1) C * r ^ n) :
theorem dist_partial_sum_le_of_le_geometric {α : Type u_1} {r : } {C : } {f : α} (hf : ∀ (n : ), f n C * r ^ n) (n : ) :
dist (Finset.sum () 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} {r : } {C : } {f : α} (hr : r < 1) (hf : ∀ (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} {r : } {C : } {f : α} (hr : r < 1) (hf : ∀ (n : ), f n C * r ^ n) {a : α} (ha : HasSum f a) (n : ) :
(Finset.sum () 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} (u : α) (n : ) :
dist (Finset.sum (Finset.range (n + 1)) fun k => u k) (Finset.sum () fun k => u k) = u n
@[simp]
theorem dist_partial_sum' {α : Type u_1} (u : α) (n : ) :
dist (Finset.sum () 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} {C : } {u : α} {r : } (hr : r < 1) (h : ∀ (n : ), u n C * r ^ n) :
CauchySeq fun n => Finset.sum () fun k => u k
theorem NormedAddCommGroup.cauchy_series_of_le_geometric' {α : Type u_1} {C : } {u : α} {r : } (hr : r < 1) (h : ∀ (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} {C : } {u : α} {N : } {r : } (hr₀ : 0 < r) (hr₁ : r < 1) (h : ∀ (n : ), n Nu 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} [] [] (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} [] [] (x : R) (h : x < 1) :
∑' (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} [] [] (x : R) (h : x < 1) :
(∑' (i : ), x ^ i) * (1 - x) = 1
theorem mul_neg_geom_series {R : Type u_4} [] [] (x : R) (h : x < 1) :
(1 - x) * ∑' (i : ), x ^ i = 1

### Summability tests based on comparison with geometric series #

theorem summable_of_ratio_norm_eventually_le {α : Type u_4} [] {f : α} {r : } (hr₁ : r < 1) (h : ∀ᶠ (n : ) in Filter.atTop, f (n + 1) r * f n) :
theorem summable_of_ratio_test_tendsto_lt_one {α : Type u_4} [] {f : α} {l : } (hl₁ : l < 1) (hf : ∀ᶠ (n : ) in Filter.atTop, f n 0) (h : Filter.Tendsto (fun n => f (n + 1) / f n) Filter.atTop (nhds l)) :
theorem not_summable_of_ratio_norm_eventually_ge {α : Type u_4} {f : α} {r : } (hr : 1 < r) (hf : ∃ᶠ (n : ) in Filter.atTop, f n 0) (h : ∀ᶠ (n : ) in Filter.atTop, r * f n f (n + 1)) :
theorem not_summable_of_ratio_test_tendsto_gt_one {α : Type u_4} {f : α} {l : } (hl : 1 < l) (h : Filter.Tendsto (fun n => f (n + 1) / f n) Filter.atTop (nhds l)) :

### Dirichlet and alternating series tests #

theorem Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded {E : Type u_4} [] {b : } {f : } {z : E} (hfa : ) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) (hgb : ∀ (n : ), Finset.sum () 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} [] {b : } {f : } {z : E} (hfa : ) (hf0 : Filter.Tendsto f Filter.atTop (nhds 0)) (hzb : ∀ (n : ), Finset.sum () 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 : ) :
Finset.sum () fun i => (-1) ^ i 1
theorem Monotone.cauchySeq_alternating_series_of_tendsto_zero {f : } (hfa : ) (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 : } (hfa : ) (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 : } (hfa : ) (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 : } (hfa : ) (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 : ) :
Summable fun n => x ^ 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 : ) :
Filter.Tendsto (fun n => x ^ n / ↑()) Filter.atTop (nhds 0)