lean_workbook_plus_21784

⚠️ 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.

Determine the convergence of the series $\sum_{j = 1}^{\infty} \frac{1}{j^{( 1 + \frac{1}{j} )}}$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_21784 : ∀ j : ℕ, j > 0 → 0 < (1 / j ^ (1 + 1 / j))   :=  by sorry

The formal statement changes or incorrectly supplies part of the source problem; the accepted note identifies a concrete witness or definitional mechanism.

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

Proposed corrected statement

Define the real series ∑_{j≥1}1/j^(1+1/j) using Real.rpow and real division, and prove its partial sums tend to +∞ (hence it is not summable). A natural-number termwise positivity assertion does not formalize convergence.

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.