The Ramanujan Machine project has catalogued many conjectural continued fraction identities involving zeta constants. In this paper we present a unified construction that produces a family of continued fractions whose limits are rational combinations of zeta values. For any integer m 2 and a suitably chosen polynomial R, we obtain a continued fraction with partial numerators bₙ=-n^2m and partial denominators aₙ given by a simple rule, and we prove that its value is \ (1R (0) +₊=₁^ 1 (k+1) ᵐ R (k) R (k-1) ) ^-1. \ By selecting R appropriately, this family produces closed forms such as \ -1 (4) +4 (2) -8, 22 (5) +6 (3) -9, 22 (5) -2 (3) +1. proof uses an explicit factorial--polynomial closed form for the numerator sequence, a telescoping difference equation, and elementary partial fraction techniques. The results confirm several identities proposed by the Ramanujan Machine (after correcting suspected typographical errors in the original conjectures). We also present a natural one‑parameter extension that yields families depending on an additional scale parameter c, which contains the original family as the special case c=1.
Lezhe Gao (Wed,) studied this question.