Limits of sequences
Problem 7.552 · easy
Write the first four terms of \( \displaystyle a_n = \frac{2^{n}}{n!} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}2\\2\\\frac{4}{3}\\\frac{2}{3}\end{matrix}\right] \]The first four terms.✓ Proved
- Factorials beat exponentials.
- \[ \lim_{n \to \infty}\left(\frac{2^{n}}{n!}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 2, 2, \frac{4}{3}, \frac{2}{3};\ 0 \)
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 informal and does not provide a rigorous justification for the convergence of the sequence. It would be clearer to reference a specific theorem (e.g., ratio test or Stirling’s approximation) to support the conclusion.qwen3.6:27b-mlx: fail (error) — The solution fails to compute the first term a_1 = 2^1/1! = 2 correctly in the context of the sequence definition if n starts at 1, but more critically, it lists a_1=2, a_2=2, a_3=4/3, a_4=2/3. Let's check: a_1 = 2/1 = 2. a_2 = 4/2 = 2. a_3 = 8/6 = 4/3. a_4 = 16/24 = 2/3. The terms are correct. However, the justification 'Factorials beat exponentials' is a heuristic, not a rigorous proof, but often accepted in introductory contexts. The main issue is usually whether the sequence starts at n=0 or n=1. If n=0, a_0 = 1. The problem asks for 'the first four terms'. Usually, this implies n=1, 2, 3, 4 unless specified otherwise. The provided terms correspond to n=1..4. The limit is 0. The reasoning is weak but the result is correct. Wait, looking closer at the prompt's constraints: 'Check that... each conclusion follows from the lines before it'. The sentence 'Factorials beat exponentials' is a hand-wavy justification for the limit. Is it an error? It's not a rigorous proof. But is it 'wrong'? In many calculus courses, this intuition is accepted. However, the prompt asks to find what is wrong. Let's look for a harder error. Did it calculate the terms right? Yes. Did it find the limit right? Yes. Is the justification sufficient? It's informal. But is there a factual error? No. Let's re-read the terms. a_1=2, a_2=2, a_3=4/3, a_4=2/3. Correct. Limit 0. Correct. Why would this fail? Perhaps the 'first four terms' implies n=0,1,2,3? If n=0, a_0=1, a_1=2, a_2=2, a_3=4/3. The solution gives 2, 2, 4/3, 2/3. This corresponds to n=1,2,3,4. Standard convention for 'sequence a_n' without specified start is often n=1. If the problem implied n=0, the answer is wrong. Without specification, n=1 is standard. Let's look at the justification again. 'Factorials beat exponentials' is not a theorem. It's a heuristic. A rigorous solution would use the Ratio Test or squeeze theorem. Is this 'misleading'? It teaches a student that heuristics are proofs. That is misleading. Verdict: fail, severity: misleading.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-13 — The sentence "Factorials beat exponentials" is informal and does not provide a rigorous justification for the convergence of the sequence. It would be clearer to reference a specific theorem (e.g., ratio test or Stirling’s approximation) to support the conclusion.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to compute the first term a_1 = 2^1/1! = 2 correctly in the context of the sequence definition if n starts at 1, but more critically, it lists a_1=2, a_2=2, a_3=4/3, a_4=2/3. Let's check: a_1 = 2/1 = 2. a_2 = 4/2 = 2. a_3 = 8/6 = 4/3. a_4 = 16/24 = 2/3. The terms are correct. However, the justification 'Factorials beat exponentials' is a heuristic, not a rigorous proof, but often accepted in introductory contexts. The main issue is usually whether the sequence starts at n=0 or n=1. If n=0, a_0 = 1. The problem asks for 'the first four terms'. Usually, this implies n=1, 2, 3, 4 unless specified otherwise. The provided terms correspond to n=1..4. The limit is 0. The reasoning is weak but the result is correct. Wait, looking closer at the prompt's constraints: 'Check that... each conclusion follows from the lines before it'. The sentence 'Factorials beat exponentials' is a hand-wavy justification for the limit. Is it an error? It's not a rigorous proof. But is it 'wrong'? In many calculus courses, this intuition is accepted. However, the prompt asks to find what is wrong. Let's look for a harder error. Did it calculate the terms right? Yes. Did it find the limit right? Yes. Is the justification sufficient? It's informal. But is there a factual error? No. Let's re-read the terms. a_1=2, a_2=2, a_3=4/3, a_4=2/3. Correct. Limit 0. Correct. Why would this fail? Perhaps the 'first four terms' implies n=0,1,2,3? If n=0, a_0=1, a_1=2, a_2=2, a_3=4/3. The solution gives 2, 2, 4/3, 2/3. This corresponds to n=1,2,3,4. Standard convention for 'sequence a_n' without specified start is often n=1. If the problem implied n=0, the answer is wrong. Without specification, n=1 is standard. Let's look at the justification again. 'Factorials beat exponentials' is not a theorem. It's a heuristic. A rigorous solution would use the Ratio Test or squeeze theorem. Is this 'misleading'? It teaches a student that heuristics are proofs. That is misleading. Verdict: fail, severity: misleading.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution claims the first four terms are a_1 through a_4, but the standard convention for sequences is to start at n=0 unless specified otherwise. Even if starting at n=1, the problem asks for 'the first four terms', which usually implies n=0, 1, 2, 3 or n=1, 2, 3, 4. The provided terms 2, 2, 4/3, 2/3 correspond to n=1, 2, 3, 4. However, the term for n=0 is 2^0/0! = 1. If the sequence starts at n=0, the first four terms are 1, 2, 2, 4/3. If it starts at n=1, the terms are correct. But more critically, the sentence 'Factorials beat exponentials' is a heuristic, not a rigorous justification for convergence, though the limit calculation is correct. The main issue is the ambiguity of the starting index. Usually, 'first four terms' implies n=0, 1, 2, 3. Let's check n=0: a_0 = 1. n=1: a_1 = 2. n=2: a_2 = 2. n=3: a_3 = 4/3. The solution lists 2, 2, 4/3, 2/3. This corresponds to n=1, 2, 3, 4. If the problem implies n starts at 1, it's fine. If n starts at 0, it's wrong. Without explicit instruction, n=0 is standard for factorials. However, the bigger error is likely the lack of rigor in the convergence argument, but the prompt says 'determine whether... find its limit'. The limit is 0. The terms are the main potential error. Let's assume standard n=0 start. Then the terms are wrong. If n=1 start, they are right. Given the ambiguity, listing n=1..4 is a specific choice. But wait, looking at the terms: 2, 2, 4/3, 2/3. a_1=2, a_2=2, a_3=4/3, a_4=2/3. This is consistent. Is it wrong? Many textbooks start sequences at n=1. However, the phrase 'Factorials beat exponentials' is informal. The real issue might be that the solution doesn't explicitly state the starting index. But is it an 'error'? If the standard is n=0, yes. If n=1, no. Let's look at the limit part. Limit is 0. Correct. The sentence 'Factorials beat exponentials' is a bit hand-wavy but captures the intuition. The most likely 'error' expected in these adversarial reviews is often the starting index or a missing justification. Let's look closer. a_0 = 1. a_1 = 2. a_2 = 2. a_3 = 4/3. The solution gives 2, 2, 4/3, 2/3. This is a_1, a_2, a_3, a_4. If the problem meant n=0, 1, 2, 3, the answer is wrong. If the problem meant n=1, 2, 3, 4, the answer is right. In calculus contexts involving n!, n=0 is very common. In sequences, n=1 is also common. However, the solution does not specify. This is ambiguous. But is it an error? Let's check the limit justification. 'Factorials beat exponentials' is not a theorem. It's a heuristic. A student might learn that this phrase is sufficient proof, which is false. This is 'misleading'. But is it an 'error'? The verdict 'fail' with 'misleading' fits. Or 'error' if the terms are wrong. Let's assume n=0 is standard. Then terms are wrong. Let's assume n=1 is standard. Then terms are right. I will flag the heuristic as misleading/informal and the potential index error. Actually, the most robust critique is that 'Factorials beat exponentials' is not a valid mathematical argument for convergence, it's a mnemonic. The limit calculation is done by CAS, so that's fine. The sentence is the problem. It teaches a student that a slogan is a proof. That is misleading. Also, if n=0 is intended, the terms are wrong. I'll go with error due to likely wrong terms (n=0 is standard for factorials) and misleading heuristic.gpt-oss:20b: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.