log_pos_real' — Mathlib

English

For a ≥ 0, log(1 + a) equals the series with terms 2/(2k+1) (a/(a+2))^{2k+1}.

Русский

Для a ≥ 0 логарифм log(1 + a) равен сумме с членами 2/(2k+1) (a/(a+2))^{2k+1}.

LaTeX

$$$\sum_{k=0}^{\infty} 2 \frac{1}{2k+1} \left(\frac{a}{a+2}\right)^{2k+1} = \log(1 + a), \quad a \ge 0$$$

Lean4

theorem log_pos_real' {x : ℝ≥0∞} (h : 0 < x.toReal) : log x = Real.log (ENNReal.toReal x) := by
  simp [log, (ENNReal.toReal_pos_iff.1 h).1.ne', (ENNReal.toReal_pos_iff.1 h).2.ne]

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