A family of equivalent representations of ln 2 from the fixed-point equation 1 − b = 2b^(2m+2)

The condition that the partial sum of the first 2m+2 terms of the alternating geometric series equals ½ reduces to the fixed-point equation 1 − b = 2b²ᵐ⁺². Under the substitution b = e⁻ᵗ, this becomes the transcendental family eᵗ − 1 = 2e⁻⁽²ᵐ⁺¹⁾ᵗ, whose unique positive roots t*(m) satisfy the algebraic identity ln 2 = (2m+1)t*(m) + ln t*(m) − ln Td(t*(m)), where Td(t) = t/(eᵗ − 1) is the Todd function. Since t*(m) ≤ ln 2 < 2π for all m ≥ 0, the Taylor expansion of Td at the root is absolutely convergent and yields a family of equivalent representations of ln 2 parametrised by the Bernoulli numbers. The note records these identities, gives a Lambert W asymptotic for t*(m) with a numerical table (m = 0,…,4), and observes two Dirichlet series over the index set of positive odd integers 2m+1: the alternating sum recovers L(s, χ₄) and the unsigned sum gives (1 − 2⁻ˢ)ζ(s). No claim is made beyond these elementary equivalences.