Limits of sequences
Problem 7.504 · easy
Write the first four terms of \( \displaystyle a_n = \frac{5 n - 3}{2 n^{2} + 1} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}\frac{2}{3}\\\frac{7}{9}\\\frac{12}{19}\\\frac{17}{33}\end{matrix}\right] \]The first four terms.✓ Proved
- The denominator has the higher degree.
- \[ \lim_{n \to \infty}\left(\frac{5 n - 3}{2 n^{2} + 1}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = \frac{2}{3}, \frac{7}{9}, \frac{12}{19}, \frac{17}{33};\ 0 \)
Lines: 2 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | 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: fail (misleading) — The sentence 'The denominator has the higher degree' is an informal heuristic, not a rigorous justification for the limit. A student should be taught to apply the formal limit laws (e.g., dividing by the highest power of n) rather than relying on degree comparison as a standalone proof step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (misleading) 2026-10-11 — The sentence 'The denominator has the higher degree' is an informal heuristic, not a rigorous justification for the limit. A student should be taught to apply the formal limit laws (e.g., dividing by the highest power of n) rather than relying on degree comparison as a standalone proof step.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (misleading) 2026-10-11 — The sentence 'The denominator has the higher degree' is an incomplete justification for the limit. It should explicitly state that for a rational function, if the degree of the denominator is greater than the degree of the numerator, the limit as n approaches infinity is 0.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/sequence_limits, checked 2026-10-11 with SymPy 1.14.0.