lean_workbook_plus_43538

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Prove that $\sum_{k=0}^\infty \binom{3k}{k}\dfrac{9k^2-3k-1}{(3k-1)(3k-2)}\left(\dfrac{2}{27}\right)^k = \dfrac{1}{4}.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_43538 : ∑' k : ℕ, (3 * k).choose k * (9 * k ^ 2 - 3 * k - 1) / ((3 * k - 1) * (3 * k - 2)) * (2 / 27) ^ k = 1 / 4   :=  by sorry

Natural arithmetic changes negative polynomial terms at k=0 and makes the geometric ratio2/27 and RHS1/4 zero, yielding the zero series.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Cast k and the binomial coefficient into ℝ before polynomial subtraction/division. Prove HasSum for the displayed genuine real summand with value1/4, retaining the k=0 term (which equals−1/2).

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.