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Absolute and conditional convergence

Problem 7.248 · medium

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} \left(n + 1\right)}{n^{3}} \) converge absolutely, converge conditionally, or diverge?

\[ \lim_{n \to \infty}\left(\frac{n + 1}{n^{3}}\right) = 0 \]
bₙ → 0.✓ Proved
bₙ is eventually decreasing, so the alternating series test gives convergence.
\[ \lim_{n \to \infty}\left(\frac{n + 1}{n}\right) = 1 \]
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: pass
qwen3.6:27b-mlx: fail (error) — The limit comparison test in step 3 is applied incorrectly: the limit of |a_n| / (1/n^2) is 1, but the text claims comparison with 1/n (or implies it via the limit shown). Furthermore, the solution fails to explicitly state that absolute convergence implies convergence, making the logical flow disjointed.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The limit comparison test in step 3 is applied incorrectly: the limit of |a_n| / (1/n^2) is 1, but the text claims comparison with 1/n (or implies it via the limit shown). Furthermore, the solution fails to explicitly state that absolute convergence implies convergence, making the logical flow disjointed.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The limit comparison test is applied incorrectly: the limit of |a_n| divided by 1/n^2 is 1, but the solution text claims the limit is of (n+1)/n, which is unrelated to the comparison with 1/n^2. Furthermore, the limit shown in line 3 is for (n+1)/n, not (n+1)/n^3, making the justification for absolute convergence mathematically incoherent.
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.