Limits of sequences
Problem 7.294 · easy
Write the first four terms of \( \displaystyle a_n = \frac{n - 3}{3 n^{2} + 1} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}- \frac{1}{2}\\- \frac{1}{13}\\0\\1 \cdot \frac{1}{49}\end{matrix}\right] = \left[\begin{matrix}- \frac{1}{2}\\- \frac{1}{13}\\0\\\frac{1}{49}\end{matrix}\right] \]The first four terms.✓ Proved
- The denominator has the higher degree.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{n - 3}{3 n^{2} + 1}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = - \frac{1}{2}, - \frac{1}{13}, 0, \frac{1}{49};\ 0 \)
✓ Nihil obstat Lines: 2 proved, 1 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 | ✓ 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 | the terms evaluated at n = 10⁶, 10⁶ + 1 and 10⁷ |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the first four terms and applies the standard heuristic for rational sequences (degree of denominator > degree of numerator implies limit 0). The reasoning is sound and sufficient for the context.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the first four terms and applies the standard heuristic for rational sequences (degree of denominator > degree of numerator implies limit 0). The reasoning is sound and sufficient for the context.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the first four terms and applies the standard heuristic for limits of rational functions (comparing degrees) to find the limit. The reasoning is sound and the conclusion is correct.gpt-oss:20b: pass 2026-10-05
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/sequence_limits, checked 2026-10-05 with SymPy 1.14.0.