Home›Calculus 2›Absolute and conditional convergence›Problem 7.241

Absolute and conditional convergence

Problem 7.241 · medium

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{2} + 1} \) converge absolutely, converge conditionally, or diverge?

\[ \lim_{n \to \infty} \frac{1}{n^{2} + 1} = 0 \]
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^2: a p-series that converges.✓ Proved

Answer \( \text{converges absolutely} \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies absolute convergence via limit comparison with a convergent p-series. The mention of the alternating series test is redundant but not incorrect, as absolute convergence implies conditional convergence.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies absolute convergence via limit comparison with a convergent p-series. The mention of the alternating series test is redundant but not incorrect, as absolute convergence implies conditional convergence.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The solution claims to use the Limit Comparison Test but provides the limit of the ratio |a_n| / (1/n^2) as 1, which is actually the limit of the term itself multiplied by n^2, not the ratio of the series terms. Furthermore, it mixes up the Alternating Series Test (for conditional convergence) and the Limit Comparison Test (for absolute convergence) in a confusing manner, failing to explicitly state that absolute convergence implies convergence.
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.