We provide a complete proof for a conjectured continued fraction identity catalogued by the Ramanujan Machine project. The continued fraction is defined by\ (a₀ = 4, \; aₙ = 3n²+7n+4\; (n 1), \; bₙ = -2n² (n+1) ²\; (n 1), \) and we prove that its limit is exactly \ (11- 2\). The proof establishes a closed form for the numerators \ (Pₙ\), expresses the convergents via a telescoping sum, and evaluates the resulting infinite series using elementary integration. Convergence follows directly from the explicit expression of the convergents as partial sums of a convergent series.
Gao Lezhe (Sat,) studied this question.