Theorems · Theorem · complex analysis

FormalMultilinearSeries.le_radius_of_isBigO

∀ {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  (p : FormalMultilinearSeries 𝕜 E F) {r : NNReal},
  ((fun n => ‖p n‖ * ↑r ^ n) =O[Filter.atTop] fun x => 1) → ↑r ≤ p.radius

If ‖pₙ‖ rⁿ = O(1), as n → ∞, then the radius of p is at least r.

Defined in

Mathlib.Analysis.Analytic.ConvergenceRadius

Cited by

7 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace