normThreshold_eq — Mathlib

English

The explicit closed form of the threshold sequence is normThreshold(a,n) = a(1+√2)(√2^{n+1} − 1) for all n.

Русский

Явное closed form выражение порога: normThreshold(a,n) = a(1+√2)(√2^{n+1} − 1) для всех n.

LaTeX

$$$\\operatorname{normThreshold}(a,n) = a\\,(1+\\sqrt{2})\\,(\\sqrt{2}^{\\;n+1}-1)$$$

Lean4

theorem normThreshold_eq (n : ℕ) : normThreshold a n = a * (1 + √2) * (√2 ^ (n + 1) - 1) :=
  by
  rw [normThreshold, arithGeom_same_eq_mul_div (by simp), div_eq_mul_inv, Real.inv_sqrt_two_sub_one]
  ring

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