Absolute and conditional convergence
Problem 7.244 · medium
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{\sqrt{n}} \) converge absolutely, converge conditionally, or diverge?
- \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0 \]bₙ → 0.✓ Proved
- bₙ is eventually decreasing, so the alternating series test gives convergence.
- \[ \lim_{n \to \infty} 1 = 1 \]Limit comparison of |aₙ| with 1/n^1/2: a p-series that diverges.✓ Proved
Answer \( \text{converges conditionally} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's Sum.is_convergent on the series and on its absolute values |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution claims 'Limit comparison of |aₙ| with 1/n^1/2' but provides the limit of the constant 1, not the limit of the ratio |aₙ| / (1/√n). While the limit of the ratio is indeed 1, the equation shown (Limit(1, ...)) does not represent the comparison test calculation and is mathematically disconnected from the claim.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims 'Limit comparison of |aₙ| with 1/n^1/2' but provides the limit of the constant 1, not the limit of the ratio |aₙ| / (1/√n). While the limit of the ratio is indeed 1, the equation shown (Limit(1, ...)) does not represent the comparison test calculation and is mathematically disconnected from the claim.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims to use limit comparison with 1/n^(1/2) but computes the limit of 1, which is incorrect. The limit of |a_n| / (1/n^(1/2)) is 1, but the text says 'Limit(1, n, oo) = 1', implying the comparison term was 1, not 1/sqrt(n). This is a factual error in the justification for divergence of the absolute series.gpt-oss:20b: pass 2026-10-04
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/alternating_series, checked 2026-10-04 with SymPy 1.14.0.