lean_workbook_plus_7346

⚠️ Retired — mistyped fractional exponent

The exponent ^(3 / 2) in the Lean statement below is natural-number division, not a real exponent, so it collapses to an integer power and the displayed radical is not what is being asserted. Do not import this node or use it as a dependency.

Determine the convergence of the series $\sum_{n\ge 1}\frac{((-1)^n\sqrt{n}+1)}{n^{3/2}}$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_7346 : ∀ n : ℕ, n ≥ 1 → (‖((-1 : ℝ)^n * Real.sqrt n + 1) / n^(3 / 2)‖ < 1 / n)   :=  by sorry

A numeric-literal exponent carrying no type ascription is elaborated at type ℕ (via HPow ℝ ℕ ℝ, Monoid.npow), and natural-number division truncates. Each occurrence therefore collapses:

• ^(3 / 2) → ^1, i.e. the 3/2 power is replaced by the 1th power

The recorded counterexample refutes the collapsed statement. It says nothing about the exercise displayed on this page, which is a different proposition.

What a faithful statement would require

Every fractional exponent must be given a real type, e.g. x ^ ((1 : ℝ) / 3) using Real.rpow, or be written with the intended root operation (Real.sqrt, or the signed real cube root where the radicand may be negative). Real powers also need their own domain hypotheses: rpow is only the intended root for a nonnegative base.

Proposed corrected statement

With real exponent3/2, determine ordered convergence of∑_{n≥1}((−1)ⁿ√n+1)/n^(3/2); split it as alternating harmonic plus a3/2 p-series. It converges conditionally, so use ordered partial sums rather than an unconditional tsum identity.

From the public Prove2Me statement audit (wamlat/prove2me-errors). That proposal is natural-language mathematics and is not Lean-verified; it is a specification for a corrected node, not a drop-in replacement.