Limits of sequences
Problem 7.442 · easy
Write the first four terms of \( \displaystyle a_n = \frac{1}{n!} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}1\\1 \cdot \frac{1}{2}\\1 \cdot \frac{1}{6}\\1 \cdot \frac{1}{24}\end{matrix}\right] = \left[\begin{matrix}1\\\frac{1}{2}\\\frac{1}{6}\\\frac{1}{24}\end{matrix}\right] \]The first four terms.✓ Proved
- Factorials beat exponentials.
- \[ \lim_{n \to \infty} \frac{1}{n!} = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24};\ 0 \)
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 irrelevant to the problem and does not support the conclusion; it should be removed or replaced with a precise justification of convergence.qwen3.6:27b-mlx: fail (error) — The solution claims the first term is 1, implying the sequence starts at n=1, but the standard definition of n! usually implies n starts at 0 or 1. If n starts at 0, a_0 = 1. If n starts at 1, a_1 = 1. However, the phrase 'Factorials beat exponentials' is a heuristic for comparing growth rates of functions like n! vs c^n, not a rigorous justification for the limit of 1/n!. While the limit is correct, the reasoning provided is informal and technically insufficient for a calculus proof, which should invoke the Squeeze Theorem or the definition of convergence. More critically, if the problem implies n starts at 1 (as is common in sequences unless specified otherwise), the terms are correct. But the justification 'Factorials beat exponentials' is misleading because 1/n! goes to 0 because n! grows without bound, not specifically because it beats an exponential (though it does). A better justification is that n! -> infinity. The main error is the lack of rigorous justification for the limit.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-08 — The sentence "Factorials beat exponentials" is irrelevant to the problem and does not support the conclusion; it should be removed or replaced with a precise justification of convergence.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution claims the first term is 1, implying the sequence starts at n=1, but the standard definition of n! usually implies n starts at 0 or 1. If n starts at 0, a_0 = 1. If n starts at 1, a_1 = 1. However, the phrase 'Factorials beat exponentials' is a heuristic for comparing growth rates of functions like n! vs c^n, not a rigorous justification for the limit of 1/n!. While the limit is correct, the reasoning provided is informal and technically insufficient for a calculus proof, which should invoke the Squeeze Theorem or the definition of convergence. More critically, if the problem implies n starts at 1 (as is common in sequences unless specified otherwise), the terms are correct. But the justification 'Factorials beat exponentials' is misleading because 1/n! goes to 0 because n! grows without bound, not specifically because it beats an exponential (though it does). A better justification is that n! -> infinity. The main error is the lack of rigorous justification for the limit.qwen3.6:27b-mlx: fail (misleading) 2026-10-08 — The phrase 'Factorials beat exponentials' is a heuristic for comparing growth rates, not a rigorous justification for the limit of 1/n!. It fails to address the specific sequence and could mislead students into thinking this comparison is the standard proof method for this limit.gpt-oss:20b: pass 2026-10-08
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.