Comparison and limit comparison tests
Problem 7.249 · easy
Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n^{2} + 4 n + 4}{3 n^{4} + 4 n^{3} + 3 n^{2} + 2 n + 1} \) converges or diverges.
- For large n the term behaves like 1/n^2, so compare with that p-series.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{n^{2} \left(4 n^{2} + 4 n + 4\right)}{3 n^{4} + 4 n^{3} + 3 n^{2} + 2 n + 1}\right) = \frac{4}{3} \]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.Reviewed
Answer \( \text{converges} \)
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies the Limit Comparison Test with the p-series 1/n^2. The limit calculation is correct, and the conclusion follows logically.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Limit Comparison Test with the p-series 1/n^2. The limit calculation is correct, and the conclusion follows logically.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Limit Comparison Test with the p-series 1/n^2. The limit calculation is correct, and the conclusion follows logically.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/comparison_tests, checked 2026-10-04 with SymPy 1.14.0.