Comparison and limit comparison tests
Problem 7.413 · easy
Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{3}{4 n + 3} \) converges or diverges.
- For large n the term behaves like 1/n^1, so compare with that p-series.
- \[ \lim_{n \to \infty}\left(\frac{3 n}{4 n + 3}\right) = \frac{3}{4} \]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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly computes the limit of the ratio of terms. For the Limit Comparison Test with b_n = 1/n, the limit should be lim (a_n / b_n)
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly computes the limit of the ratio of terms. For the Limit Comparison Test with b_n = 1/n, the limit should be lim (a_n / b_n)gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The limit comparison is performed with the wrong series. The text claims to compare with 1/n, but the equation calculates the limit of the original term (3/(4n+3)), not the ratio of the terms (which would be 3/4). To use Limit Comparison Test with 1/n, one must compute lim (a_n / b_n). The provided equation computes lim a_n, which is 0, not 3/4. Even if the limit value 3/4 is correct for the ratio, the equation shown is mathematically incorrect for the stated purpose.
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 2026-10-07 with SymPy 1.14.0.