Theorems · Theorem · several complex variables

HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ENNReal} {y : E},
  HasFPowerSeriesWithinOnBall f p s x r →
    y ∈ Metric.eball 0 r →
      x + y ∈ insert x s → Filter.Tendsto (fun z => p.partialSum z.1 z.2) (Filter.atTop ×ˢ nhds y) (nhds (f (x + y)))

If a function admits a power series expansion within a ball, then the partial sums p.partialSum n z converge to f (x + y) as n → ∞ and z → y. Note that x + z doesn't need to belong to the set where the power series expansion holds.

Defined in

Mathlib.Analysis.Analytic.Basic

Cited by

2 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace