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Comparison and limit comparison tests

Problem 7.252 · easy

Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{3}{4 n + 2} \) converges or diverges.

For large n the term behaves like 1/n^1, so compare with that p-series.
\[ \lim_{n \to \infty}\left(\frac{3 n}{4 n + 2}\right) = \frac{3}{4} \]
The limit is finite and positive, so the two series do the same thing.✓ Proved
Σ 1/n^1 diverges (p ≤ 1), so the given series diverges.

Answer \( \text{diverges} \)

Lines: 1 proved, 2 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: fail (style) — [domain objection, downgraded to style] The solution applies the Limit Comparison Test but computes the limit of the term ratio (a_n/b_n) instead of the limit of the terms themselves or the ratio of the series terms correctly. Specifically, line 2 computes lim (3n / (4n+2)) which is not the standard form for LCT comparison with 1/n (which would be lim (3/(4n+2)) / (1/n) = 3/4). While the numerical result 3/4 is coincidentally correct for the ratio a_n/b_n if b_n=1/n, the expression written '3*n/(4*n + 2)' is ambiguous and technically represents n*a_n, not a_n/b_n. More importantly, the sentence in step 1 says 'compare with that p-series' but step 2's equation is labeled as a limit of a single expression rather than a ratio of two series terms, making the application of the theorem unclear and potentially confusing to a student. However, the biggest issue is that the equation shown is `Limit(3*n/(4*n + 2), n, oo, dir='-')`. If we are comparing `a_n = 3/(4n+2)` with `b_n = 1/n`, the limit should be `lim (a_n/b_n) = lim (3/(4n+2) * n/1) = lim (3n/(4n+2))`. So the equation is actually correct for the LCT ratio. Wait, let me re-read. `a_n = 3/(4n+2)`. `b_n = 1/n`. `a_n/b_n = (3/(4n+2)) / (1/n) = 3n/(4n+2)`. The limit is 3/4. This is correct. Why did I think it was wrong? Ah, the prompt says 'Limit(3*n/(4*n + 2)...'. This IS the ratio. So the math is correct. Is there a subtle error? 'For large n the term behaves like 1/n^1'. This is heuristic. The LCT is rigorous. The conclusion is correct. Is it misleading? No. Is it an error? No. Let me look closer. 'Limit(3*n/(4*n + 2), n, oo, dir='-')'. The `dir='-'` usually means approaching from the left, but n goes to infinity. This is a syntax artifact of the CAS, not a mathematical error in the solution text. The solution is sound. Wait, look at step 1: 'behaves like 1/n^1'. Step 2: Limit is 3/4. Step 3: Diverges. This is a standard LCT application. It seems correct. Why would it fail? Maybe the 'dir' parameter is confusing? No, that's CAS output. Maybe the comparison series isn't explicitly defined as b_n = 1/n in the text, only implied? Step 1 says 'compare with that p-series'. Step 2 shows the limit of the ratio. Step 3 concludes. It's a bit terse but correct. Let me check if there's a stricter interpretation. 'Limit(3*n/(4*n + 2)...' is the limit of n*a_n. If b_n = 1/n, then a_n/b_n = n*a_n. So the limit is indeed 3/4. The logic holds. Verdict: pass.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution applies the Limit Comparison Test but computes the limit of the term ratio (a_n/b_n) instead of the limit of the terms themselves or the ratio of the series terms correctly. Specifically, line 2 computes lim (3n / (4n+2)) which is not the standard form for LCT comparison with 1/n (which would be lim (3/(4n+2)) / (1/n) = 3/4). While the numerical result 3/4 is coincidentally correct for the ratio a_n/b_n if b_n=1/n, the expression written '3*n/(4*n + 2)' is ambiguous and technically represents n*a_n, not a_n/b_n. More importantly, the sentence in step 1 says 'compare with that p-series' but step 2's equation is labeled as a limit of a single expression rather than a ratio of two series terms, making the application of the theorem unclear and potentially confusing to a student. However, the biggest issue is that the equation shown is `Limit(3*n/(4*n + 2), n, oo, dir='-')`. If we are comparing `a_n = 3/(4n+2)` with `b_n = 1/n`, the limit should be `lim (a_n/b_n) = lim (3/(4n+2) * n/1) = lim (3n/(4n+2))`. So the equation is actually correct for the LCT ratio. Wait, let me re-read. `a_n = 3/(4n+2)`. `b_n = 1/n`. `a_n/b_n = (3/(4n+2)) / (1/n) = 3n/(4n+2)`. The limit is 3/4. This is correct. Why did I think it was wrong? Ah, the prompt says 'Limit(3*n/(4*n + 2)...'. This IS the ratio. So the math is correct. Is there a subtle error? 'For large n the term behaves like 1/n^1'. This is heuristic. The LCT is rigorous. The conclusion is correct. Is it misleading? No. Is it an error? No. Let me look closer. 'Limit(3*n/(4*n + 2), n, oo, dir='-')'. The `dir='-'` usually means approaching from the left, but n goes to infinity. This is a syntax artifact of the CAS, not a mathematical error in the solution text. The solution is sound. Wait, look at step 1: 'behaves like 1/n^1'. Step 2: Limit is 3/4. Step 3: Diverges. This is a standard LCT application. It seems correct. Why would it fail? Maybe the 'dir' parameter is confusing? No, that's CAS output. Maybe the comparison series isn't explicitly defined as b_n = 1/n in the text, only implied? Step 1 says 'compare with that p-series'. Step 2 shows the limit of the ratio. Step 3 concludes. It's a bit terse but correct. Let me check if there's a stricter interpretation. 'Limit(3*n/(4*n + 2)...' is the limit of n*a_n. If b_n = 1/n, then a_n/b_n = n*a_n. So the limit is indeed 3/4. The logic holds. Verdict: pass.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The limit comparison is performed with the wrong series. The text claims to compare with 1/n, but the equation calculates the limit of a_n / (1/n^2)
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.