File: Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean

full_name: Filter.IsBoundedUnder.isLittleO_sub_self_inv
code: /-- If `f : 𝕜 → E` is bounded in a punctured neighborhood of `a`, then `f(x) = o((x - a)⁻¹)` as
`x → a`, `x ≠ a`. -/
theorem Filter.IsBoundedUnder.isLittleO_sub_self_inv {𝕜 E : Type*} [NormedField 𝕜] [Norm E] {a : 𝕜}
    {f : 𝕜 → E} (h : IsBoundedUnder (· ≤ ·) (𝓝[≠] a) (norm ∘ f)) :
    f =o[𝓝[≠] a] fun x => (x - a)⁻¹ 
goal: f =o[𝓝[≠] a] fun x => (x - a)⁻¹

full_name: pow_div_pow_eventuallyEq_atTop
code: theorem pow_div_pow_eventuallyEq_atTop {p q : ℕ} :
    (fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atTop] fun x => x ^ ((p : ℤ) - q) 
goal: (fun x => x ^ p / x ^ q) =ᶠ[Filter.atTop] fun x => x ^ (↑p - ↑q)

full_name: pow_div_pow_eventuallyEq_atBot
code: theorem pow_div_pow_eventuallyEq_atBot {p q : ℕ} :
    (fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ ((p : ℤ) - q) 
goal: (fun x => x ^ p / x ^ q) =ᶠ[Filter.atBot] fun x => x ^ (↑p - ↑q)

full_name: tendsto_pow_div_pow_atTop_atTop
code: theorem tendsto_pow_div_pow_atTop_atTop {p q : ℕ} (hpq : q < p) :
    Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop atTop 
goal: Filter.Tendsto (fun x => x ^ p / x ^ q) Filter.atTop Filter.atTop

full_name: tendsto_pow_div_pow_atTop_zero
code: theorem tendsto_pow_div_pow_atTop_zero [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p q : ℕ}
    (hpq : p < q) : Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop (𝓝 0) 
goal: Filter.Tendsto (fun x => x ^ p / x ^ q) Filter.atTop (𝓝 0)

full_name: Asymptotics.isLittleO_pow_pow_atTop_of_lt
code: theorem Asymptotics.isLittleO_pow_pow_atTop_of_lt [OrderTopology 𝕜] {p q : ℕ} (hpq : p < q) :
    (fun x : 𝕜 => x ^ p) =o[atTop] fun x => x ^ q 
goal: (fun x => x ^ p) =o[Filter.atTop] fun x => x ^ q

full_name: Asymptotics.IsBigO.trans_tendsto_norm_atTop
code: theorem Asymptotics.IsBigO.trans_tendsto_norm_atTop {α : Type*} {u v : α → 𝕜} {l : Filter α}
    (huv : u =O[l] v) (hu : Tendsto (fun x => ‖u x‖) l atTop) :
    Tendsto (fun x => ‖v x‖) l atTop 
goal: Filter.Tendsto (fun x => ‖v x‖) l Filter.atTop

full_name: Asymptotics.IsLittleO.sum_range
code: theorem Asymptotics.IsLittleO.sum_range {α : Type*} [NormedAddCommGroup α] {f : ℕ → α} {g : ℕ → ℝ}
    (h : f =o[atTop] g) (hg : 0 ≤ g) (h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop) :
    (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i 
goal: (fun n => ∑ i ∈ Finset.range n, f i) =o[Filter.atTop] fun n => ∑ i ∈ Finset.range n, g i

full_name: Asymptotics.isLittleO_sum_range_of_tendsto_zero
code: theorem Asymptotics.isLittleO_sum_range_of_tendsto_zero {α : Type*} [NormedAddCommGroup α]
    {f : ℕ → α} (h : Tendsto f atTop (𝓝 0)) :
    (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => (n : ℝ) 
goal: (fun n => ∑ i ∈ Finset.range n, f i) =o[Filter.atTop] fun n => ↑n

full_name: Filter.Tendsto.cesaro_smul
code: /-- The Cesaro average of a converging sequence converges to the same limit. -/
theorem Filter.Tendsto.cesaro_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {u : ℕ → E}
    {l : E} (h : Tendsto u atTop (𝓝 l)) :
    Tendsto (fun n : ℕ => (n⁻¹ : ℝ) • ∑ i ∈ range n, u i) atTop (𝓝 l) 
goal: Filter.Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 l)

full_name: Filter.Tendsto.cesaro
code: /-- The Cesaro average of a converging sequence converges to the same limit. -/
theorem Filter.Tendsto.cesaro {u : ℕ → ℝ} {l : ℝ} (h : Tendsto u atTop (𝓝 l)) :
    Tendsto (fun n : ℕ => (n⁻¹ : ℝ) * ∑ i ∈ range n, u i) atTop (𝓝 l) 
goal: Filter.Tendsto (fun n => (↑n)⁻¹ * ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 l)