Comparison and limit comparison tests
Problem 7.469 · easy
Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{5}{2 n + 1} \) converges or diverges.
- For large n the term behaves like 1/n^1, so compare with that p-series.
- \[ \lim_{n \to \infty}\left(\frac{5 n}{2 n + 1}\right) = \frac{5}{2} \]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 limit comparison is performed with the wrong series. The text claims to compare with 1/n, but the equation computes the limit of the ratio with
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — 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 computes the limit of the ratio withqwen3.6:27b-mlx: fail (error) 2026-10-09 — The limit calculation is incorrect; the limit of the ratio a_n/b_n is 5/2, not 5/2 derived from 5*n/(2*n+1) which is actually 5/2, but the text says 'Limit(5*n/(2*n + 1)...' which evaluates to 5/2. Wait, 5n / (2n+1) -> 5/2. The limit is indeed 5/2. The logic is sound. Why did I think it was wrong? Ah, the prompt says 'Limit(5*n/(2*n + 1)...' = 5/2. This is correct. The comparison is with 1/n. a_n = 5/(2n+1). b_n = 1/n. a_n/b_n = 5n/(2n+1) -> 5/2. This is finite and positive. The conclusion is correct. Is there a subtle error? The sentence says 'behaves like 1/n^1'. This is informal but acceptable. The limit calculation is correct. The conclusion is correct. Wait, looking closely at the provided solution text: 'Limit(5*n/(2*n + 1), n, oo, dir='-') = 5/2'. This is the limit of the ratio. The sentence says 'compare with that p-series'. It implies the Limit Comparison Test. The hypotheses are met (positive terms). The conclusion follows. It seems correct. Let me re-read carefully. 'For large n the term behaves like 1/n^1'. a_n ~ 5/(2n) = 2.5/n. So it behaves like 1/n. The limit comparison with 1/n gives 5/2. Everything is correct. Why would it fail? Maybe the 'dir='-' is weird syntax but the result is right. Maybe the sentence 'The limit is finite and positive, so the two series do the same thing' is slightly informal but mathematically sound for LCT. I will mark it pass.gpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.