Limits of sequences
Problem 7.554 · easy
Write the first four terms of \( \displaystyle a_n = \frac{2 n - 3}{n^{2} + 1} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}- \frac{1}{2}\\1 \cdot \frac{1}{5}\\\frac{3}{10}\\\frac{5}{17}\end{matrix}\right] = \left[\begin{matrix}- \frac{1}{2}\\\frac{1}{5}\\\frac{3}{10}\\\frac{5}{17}\end{matrix}\right] \]The first four terms.✓ Proved
- The denominator has the higher degree.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{2 n - 3}{n^{2} + 1}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = - \frac{1}{2}, \frac{1}{5}, \frac{3}{10}, \frac{5}{17};\ 0 \)
✓ 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 computes the terms and identifies the limit. The reasoning that the denominator's higher degree implies a limit of 0 is standard and correct for rational functions.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the terms and identifies the limit. The reasoning that the denominator's higher degree implies a limit of 0 is standard and correct for rational functions.qwen3.6:27b-mlx: fail (style) 2026-10-13 — The solution is mathematically correct but stylistically poor. It presents the terms as a matrix equation rather than a list, and the limit justification is too brief, omitting the standard division by the highest power of n.gpt-oss:20b: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.