lean_workbook_plus_4002

⚠️ Retired — specification defect

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

For a more formal argument, look at the limsup version of the root test: \n\n $\limsup_{n\to\infty}\left[\frac{n^3(\sqrt{2}+(-1)^n)^n}{3^n}\right]^{\frac1n}=\frac{\sqrt{2}+1}3<1,$ so the series converges absolutely.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_4002 : ∀ n : ℕ, ((n^3 * (Real.sqrt 2 + (-1 : ℝ)^n)^n) / 3^n)^(1/n) < 1   :=  by sorry

An asymptotic limsup root test became a strict pointwise bound for every n, and the root exponent became natural division.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

State the genuine real-root limsup of aₙ=n³(√2+(−1)ⁿ)ⁿ/3ⁿ as(√2+1)/3<1 and conclude absolute summability. Do not require the root bound at every finite index.

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.