Absolute and conditional convergence
Problem 7.331 · medium
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{2}} \) converge absolutely, converge conditionally, or diverge?
- \[ \lim_{n \to \infty} \frac{1}{n^{2}} = 0 \]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^2: a p-series that converges.✓ Proved
Answer \( \text{converges absolutely} \)
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 fails to explicitly state that the series converges absolutely because the series of absolute values converges. While it checks for conditional convergence (AST) and absolute convergence (Limit Comparison), it does not link the result of the Limit Comparison Test to the final classification 'converges absolutely'. A student might think the AST result is the primary justification or that the two tests are unrelated. The sentence in step 2 is also slightly imprecise: b_n is decreasing for all n >= 1, not just 'eventually', though this is a minor point compared to the logical gap in the conclusion.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state that the series converges absolutely because the series of absolute values converges. While it checks for conditional convergence (AST) and absolute convergence (Limit Comparison), it does not link the result of the Limit Comparison Test to the final classification 'converges absolutely'. A student might think the AST result is the primary justification or that the two tests are unrelated. The sentence in step 2 is also slightly imprecise: b_n is decreasing for all n >= 1, not just 'eventually', though this is a minor point compared to the logical gap in the conclusion.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution claims to use limit comparison with 1/n^2 but computes the limit of 1, which is incorrect (the limit of |a_n| / (1/n^2) is 1, but the line shows Limit(1, n, oo) = 1, implying the comparison term was 1, not 1/n^2, or the calculation is garbled). Furthermore, checking absolute convergence via the p-series test is sufficient and more direct than the alternating series test, but the presentation of the limit comparison is mathematically incoherent.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.