Limits of sequences
Problem 7.503 · easy
Write the first four terms of \( \displaystyle a_n = \frac{3^{n}}{n!} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}3\\\frac{9}{2}\\\frac{9}{2}\\\frac{27}{8}\end{matrix}\right] \]The first four terms.✓ Proved
- Factorials beat exponentials.
- \[ \lim_{n \to \infty}\left(\frac{3^{n}}{n!}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 3, \frac{9}{2}, \frac{9}{2}, \frac{27}{8};\ 0 \)
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: fail (style) — The sentence "Factorials beat exponentials" is vague and does not provide a rigorous justification for the limit; a more precise statement (e.g., using the ratio test or Stirling’s approximation) would be appropriate.qwen3.6:27b-mlx: pass — The solution correctly calculates the first four terms and identifies the limit as 0. The heuristic 'Factorials beat exponentials' is a standard and correct justification for the convergence of this sequence.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the first four terms and identifies the limit as 0. The heuristic 'Factorials beat exponentials' is a standard and correct justification for the convergence of this sequence.gpt-oss:20b: fail (style) 2026-10-11 — The sentence "Factorials beat exponentials" is vague and does not provide a rigorous justification for the limit; a more precise statement (e.g., using the ratio test or Stirling’s approximation) would be appropriate.qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The problem asks for the first four terms, which typically implies starting at n=1. The solution lists a_1, a_2, a_3, a_4 correctly. However, the jugpt-oss:20b: pass 2026-10-11 — The solution correctly lists the first four terms and correctly states that the limit is 0. The informal sentence “Factorials beat exponentials” is not a formal justification but does not mislead; it could be clearer, but it does not contain an error.
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.