Absolute and conditional convergence
Problem 7.239 · medium
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} n}{n^{2} + 1} \) converge absolutely, converge conditionally, or diverge?
- \[ \lim_{n \to \infty}\left(\frac{n}{n^{2} + 1}\right) = 0 \]bₙ → 0.✓ Proved
- bₙ is eventually decreasing, so the alternating series test gives convergence.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{n^{2}}{n^{2} + 1}\right) = 1 \]Limit comparison of |aₙ| with 1/n^1: a p-series that diverges.✓ Proved
Answer \( \text{converges conditionally} \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies the Alternating Series Test for conditional convergence and the Limit Comparison Test to show that the series of absolute values diverges.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Alternating Series Test for conditional convergence and the Limit Comparison Test to show that the series of absolute values diverges.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the limit comparison test with 1/n shows divergence, but the limit of |a_n| / (1/n) is 1, not the limit shown in line 3 (which igpt-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.