Prove2Me

`BookProof.ChapterThermalTemperatureCore.geometricOccupancy_mean` (h0 : 0 ≤ r) (h1 : r < 1) : ∑' n : ℕ, (n : ℝ) * geometricOccupancy r n = r / (1 - r)

Proved
BookProof.ChapterThermalTemperatureCore.geometricOccupancy_mean

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterThermalTemperatureCore.

BookProof.ChapterThermalTemperatureCore.geometricOccupancy_mean (h0 : 0 ≤ r) (h1 : r < 1) : ∑' n : ℕ, (n : ℝ) * geometricOccupancy r n = r / (1 - r)

Formalization note: Lean 4 identifier BookProof.ChapterThermalTemperatureCore.geometricOccupancy_mean.

Preamble
-- Generated from ChapterThermalTemperatureCore.lean — theorem BookProof.ChapterThermalTemperatureCore.geometricOccupancy_mean
import Definitions.Def_ChapterCoherentOccupation
import Definitions.Def_ChapterBoseEinstein
import Mathlib
import Definitions.Def_ChapterThermalTemperatureCore
import Definitions.Def_ChapterCoherentTemperature
open BookProof.ChapterCoherentTemperature
open BookProof.ChapterThermalTemperatureCore


noncomputable section


open BookProof.ChapterCoherentTemperature BookProof.ChapterCoherentOccupation
open BookProof.ChapterBoseEinstein

variable {r : ℝ}
Formal statement
theorem BookProof.ChapterThermalTemperatureCore.geometricOccupancy_mean (h0 : 0 ≤ r) (h1 : r < 1) :
    ∑' n : ℕ, (n : ℝ) * geometricOccupancy r n = r / (1 - r) := by sorry

View graph