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Limits of sequences

Problem 7.296 · easy

Write the first four terms of \( \displaystyle a_n = \frac{5^{n}}{n!} \), and determine whether the sequence converges. If it does, find its limit.

\[ \left[\begin{matrix}5\\\frac{25}{2}\\\frac{125}{6}\\\frac{625}{24}\end{matrix}\right] \]
The first four terms.✓ Proved
Factorials beat exponentials.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{5^{n}}{n!}\right) = 0 \]
The limit.✓ Proved

Answer \( a_1, \dots, a_4 = 5, \frac{25}{2}, \frac{125}{6}, \frac{625}{24};\ 0 \)

Lines: 2 proved, 1 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the terms and identifies the limit. The heuristic 'Factorials beat exponentials' is standard and correct for this context.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the terms and identifies the limit. The heuristic 'Factorials beat exponentials' is standard and correct for this context.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to provide a rigorous justification for the limit, relying on an informal heuristic ('Factorials beat exponentials') instead of a standard test like the Ratio Test. Additionally, the limit calculation specifies a direction ('dir=-') which is irrelevant and potentially confusing for a limit at positive infinity.
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.