isEquivalent_choose — Mathlib

English

The derivative of qaryEntropy q is not continuous at p = 0.

Русский

Производная qaryEntropy q не непрерывна в p = 0.

LaTeX

$$¬ ContinuousAt (deriv (Real.qaryEntropy q)) 0$$

Lean4

/-- `n.choose k` is asymptotically equivalent to `n^k / k!`. -/
theorem isEquivalent_choose (k : ℕ) :
    (fun (n : ℕ) ↦ (n.choose k : ℝ)) ~[atTop] (fun (n : ℕ) ↦ (n ^ k / k.factorial : ℝ)) :=
  by
  conv_lhs =>
    intro n
    rw [choose_eq_descFactorial_div_factorial,
      cast_div (n.factorial_dvd_descFactorial k) (mod_cast k.factorial_ne_zero)]
  exact (isEquivalent_descFactorial k).div IsEquivalent.refl

Похожие утверждения