Series convergence tests
Problem 7.161 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{2 n + 1}{n^{2} + n + 1} \) converge or diverge?
- For large n the terms behave like 2/n^1: compare with the p-series Σ 1/n^1.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{n \left(2 n + 1\right)}{n^{2} + n + 1}\right) = 2 \]The limit is 2, positive and finite.✓ Proved
- Σ 1/n^1 diverges (p = 1), so the given series does too.Reviewed
Answer \( \text{diverges} \)
✓ 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 own convergence decision (Sum.is_convergent) agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Limit Comparison Test with the harmonic series, which is a valid method for determining divergence.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Limit Comparison Test with the harmonic series, which is a valid method for determining divergence.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Limit Comparison Test with the harmonic series. The limit calculation is correct, and the conclusion follows logically.gpt-oss:20b: pass 2026-09-27
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/series_convergence, checked 2026-09-27 with SymPy 1.14.0.