Limits of sequences
Problem 7.444 · easy
Write the first four terms of \( \displaystyle a_n = \frac{4^{n}}{n!} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}4\\8\\\frac{32}{3}\\\frac{32}{3}\end{matrix}\right] \]The first four terms.✓ Proved
- Factorials beat exponentials.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{4^{n}}{n!}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 4, 8, \frac{32}{3}, \frac{32}{3};\ 0 \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the first four terms and the limit. The heuristic 'Factorials beat exponentials' is a standard and correct justification for the convergence of n! vs a^n.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the first four terms and the limit. The heuristic 'Factorials beat exponentials' is a standard and correct justification for the convergence of n! vs a^n.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The first term of the sequence is a_1 = 4/1! = 4, but the problem asks for the first four terms, which usually implies starting from n=1. However, the provided list is 4, 8, 32/3, 32/3. Let's check: a_1=4, a_2=16/2=8, a_3=64/6=32/3, a_4=256/24=32/3. The terms are correct. The limit is 0. The reasoning 'Factorials beat exponentials' is a heuristic, not a rigorous proof, but often accepted in introductory contexts. However, looking closely at the terms: a_1=4, a_2=8, a_3=32/3 (~10.67), a_4=32/3 (~10.67). Wait, a_4 = 4^4 / 4! = 256 / 24 = 32/3. Correct. The limit is 0. The solution is technically correct in its results. Is there a subtle error? The prompt asks to 'determine whether the sequence converges'. The sentence 'Factorials beat exponentials' is a justification for the limit being 0. This is a standard heuristic. Is it misleading? It's informal but not false. Let's look for a stricter error. The problem asks for the first four terms. Usually, sequences are indexed starting at n=1 unless specified otherwise. The terms listed correspond to n=1, 2, 3, 4. The limit is 0. The solution seems correct. Why would it fail? Maybe the 'unchecked' equation label is a hint? No, I only review sentences. The sentence is 'Factorials beat exponentials'. This is a valid intuition. Is it a 'theorem'? No. Is it a 'method'? It's a heuristic. Does it follow? Yes. Is it misleading? No. Wait, look at the terms again. a_1=4, a_2=8, a_3=32/3, a_4=32/3. The sequence increases then stays same? a_5 = 4^5/120 = 1024/120 = 128/15 = 8.53. It decreases after n=3. The limit is 0. The solution is correct. I will mark it as pass.gpt-oss:20b: fail (misleading) 2026-10-08 — The sentence "Factorials beat exponentials" is an informal claim that does not provide a valid justification for the convergence of the sequence; a rigorous argument (e.g., ratio test) is required.
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 2026-10-08 with SymPy 1.14.0.