Comparison and limit comparison tests
Problem 7.518 · easy
Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{3}{4 n + 4} \) converges or diverges.
- For large n the term behaves like 1/n^1, so compare with that p-series.
- \[ \lim_{n \to \infty}\left(\frac{3 n}{4 n + 4}\right) = \frac{3}{4} \]The limit is finite and positive, so the two series do the same thing.✓ Proved
- Σ 1/n^1 diverges (p ≤ 1), so the given series diverges.
Answer \( \text{diverges} \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The limit comparison is performed with the wrong series. The solution computes the limit of the original term (which is 0) against an implicit compa
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The limit comparison is performed with the wrong series. The solution computes the limit of the original term (which is 0) against an implicit compagpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The limit comparison is performed on the wrong terms; the solution computes the limit of 3n/(4n+4) instead of the ratio of the series terms (3/(4n+4)) / (1/n). Although the final conclusion is correct, the reasoning is mathematically invalid.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.