Comparison and limit comparison tests
Problem 7.465 · easy
Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{2}{4 n^{2} + n + 2} \) converges or diverges.
- For large n the term behaves like 1/n^2, so compare with that p-series.
- \[ \lim_{n \to \infty}\left(\frac{2 n^{2}}{4 n^{2} + n + 2}\right) = \frac{1}{2} \]The limit is finite and positive, so the two series do the same thing.✓ Proved
- Σ 1/n^2 converges (p > 1), so the given series converges.
Answer \( \text{converges} \)
Lines: 1 proved, 2 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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's Sum.is_convergent |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The limit comparison test is applied with the wrong comparison term. The solution claims the term behaves like 1/n^2 but calculates the limit of a_n / (1/n^2) as 1/2, which is incorrect (the limit is 0). The correct comparison term is 1/(2n^2) or simply noting a_n ~ 1/(2n^2). Using 1/n^2 yields a limit of 0, which is inconclusive for the Limit Comparison Test unless one uses the specific variant where limit=0 implies convergence if the comparison series converges, but the text explicitly states the limit is 1/2, which is a calculation error.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The limit comparison test is applied with the wrong comparison term. The solution claims the term behaves like 1/n^2 but calculates the limit of a_n / (1/n^2) as 1/2, which is incorrect (the limit is 0). The correct comparison term is 1/(2n^2) or simply noting a_n ~ 1/(2n^2). Using 1/n^2 yields a limit of 0, which is inconclusive for the Limit Comparison Test unless one uses the specific variant where limit=0 implies convergence if the comparison series converges, but the text explicitly states the limit is 1/2, which is a calculation error.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The limit comparison is performed with the wrong series. The term behaves like 1/(2n^2), so it should be compared to 1/n^2, but the limit calculatiogpt-oss:20b: pass 2026-10-09
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/comparison_tests, checked 2026-10-09 with SymPy 1.14.0.