English
If a function has a power-series around x, the tendstoUniformlyOn' refined holds for tendstoUniformlyOn and tendstoUniformlyOn' with insertions and ball bounds.
Русский
Если функция имеет степенной ряд в точке x, то более точное условие tends toUniformlyOn' следует из tendsToUniformlyOn и области шарa.
LaTeX
$$$\text{tendstoUniformlyOn} \Rightarrow \text{tendstoUniformlyOn'}$ на соответствующих областях$$
Lean4
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop (Metric.ball (0 : E) r') :=
by
rw [← hasFPowerSeriesWithinOnBall_univ] at hf
simpa using hf.tendstoUniformlyOn h
