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Comparison and limit comparison tests

Problem 7.512 · easy

Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + n^{2} + n + 3}} \) converges or diverges.

For large n the term behaves like 1/n^3/2, so compare with that p-series.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{n^{\frac{3}{2}}}{\sqrt{n^{3} + n^{2} + n + 3}}\right) = 1 \]
The limit is finite and positive, so the two series do the same thing.✓ Proved
Σ 1/n^3/2 converges (p > 1), so the given series converges.
Reviewed

Answer \( \text{converges} \)

Lines: 1 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	SymPy's Sum.is_convergent

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Limit Comparison Test with the appropriate p-series. The logic is sound and the conclusion is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the Limit Comparison Test with the appropriate p-series. The logic is sound and the conclusion is correct.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The comparison series is identified as 1/n^(3/2), but the text writes '1/n^3/2' which is ambiguous and could be interpreted as 1/(n^3/2) or (1/n^3)/2. More critically, the limit calculation in line 2 is incorrect: the limit of n^(3/2)/sqrt(n^3 + ...) is 1, but the term in the sum is 1/sqrt(...), so the comparison should be with 1/n^(3/2). The sentence in line 1 says 'behaves like 1/n^3/2', which is syntactically confusing. However, the main error is that the limit test is applied correctly in principle, but the notation '1/n^3/2' is poor. Let's look closer. The term is a_n = 1/sqrt(n^3+...). b_n = 1/n^(3/2). lim a_n/b_n = lim n^(3/2)/sqrt(n^3+...) = 1. This is correct. The series sum 1/n^(3/2) converges because p=3/2 > 1. This is correct. Why would it fail? Ah, look at line 2: 'Limit(n**(3/2)/sqrt(n**3 + n**2 + n + 3), n, oo, dir='-') = 1'. This is correct. Line 3: 'Σ 1/n^3/2 converges'. The notation '1/n^3/2' is ambiguous. Does it mean 1/(n^(3/2)) or (1/n^3)/2? In standard order of operations, 1/n^3/2 = (1/n^3)/2 = 1/(2n^3). If it means 1/(2n^3), then p=3, which also converges. But the limit comparison was with n^(3/2) in the numerator, implying the denominator was n^(3/2). So the intended term is 1/n^(3/2). The text '1/n^3/2' is misleading/ambiguous. Is it an error? It's a notation error. But wait, look at line 1: 'behaves like 1/n^3/2'. If a student reads this as 1/(n^3/2) = 2/n^3, they might think the comparison is with 1/n^3. The limit check confirms the power is 3/2. The conclusion is correct. Is it 'error' or 'misleading'? The prompt asks to find what is wrong. The notation is sloppy. But is there a deeper error? No. The logic is sound. The verdict should probably be 'pass' if we are lenient on notation, or 'fail' with 'style' or 'misleading' if we are strict. Let's look at the options. 'error' is for wrong steps. 'misleading' is for false teaching. 'style' is for clarity. The notation '1/n^3/2' is very likely to be misread. However, in many contexts, n^3/2 is read as n^(3/2). Let's check if there is a more subtle error. The problem asks to determine convergence. The solution does so. The limit is 1. The p-series converges. Everything is correct mathematically. The only issue is the text representation of the exponent. I will mark it as 'style' or 'misleading'. Given the adversarial nature, 'misleading' is appropriate because '1/n^3/2' is not standard notation for n^(-3/2). Standard is 1/n^(3/2) or n^(-3/2). Writing 1/n^3/2 suggests (1/n^3)/2. If a student copies this notation, they might make errors later. I will go with 'misleading'.
gpt-oss:20b: pass 2026-10-11

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/comparison_tests, checked 2026-10-11 with SymPy 1.14.0.