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Comparison and limit comparison tests

Problem 7.340 · easy

Determine whether \( \displaystyle \sum_{n=1}^{\infty} \frac{4}{2 n + 4} \) converges or diverges.
  1. For large n the term behaves like 1/n^1, so compare with that p-series.
  2. \[ \lim_{n \to \infty}\left(\frac{4 n}{2 n + 4}\right) = 2 \]
    The limit is finite and positive, so the two series do the same thing.✓ Proved
  3. 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
LineStatusChecked byDetail
1Not checked—a sentence; read, not computed
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3Not checked—a sentence; read, not computed
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0SymPy's Sum.is_convergent

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: fail (error) — The Limit Comparison Test requires comparing the original term a_n = 4/(2n+4) with b_n = 1/n. The solution incorrectly computes the limit of 4n/(2n+4), which corresponds to comparing 4n with 1/n (or a_n with 1/n^2 if interpreted differently), rather than the ratio a_n/b_n = (4/(2n+4))/(1/n) = 4n/(2n+4). While the computed limit value 2 is correct for the ratio a_n/b_n, the sentence in step 1 claims the term behaves like 1/n, but step 2's equation label '4*n/(2*n + 4)' is ambiguous and technically represents the numerator of the ratio if b_n=1/n, but usually LCT is stated as lim a_n/b_n. The main issue is that step 1 says 'behaves like 1/n^1' but the equation in step 2 is just the limit of 4n/(2n+4). If b_n = 1/n, then a_n/b_n = 4n/(2n+4). The limit is 2. This is correct. However, the phrasing 'Limit(4*n/(2*n + 4)...)' is not the standard way to write the LCT limit, which is lim (a_n / b_n). It looks like the student might have confused the term itself with the ratio. But wait, if a_n = 4/(2n+4) and b_n = 1/n, then a_n/b_n = 4n/(2n+4). So the equation is correct for the LCT ratio. The conclusion is correct. Is there an error? Step 1 says 'compare with that p-series'. Step 2 calculates the limit of the ratio. Step 3 concludes divergence. This seems correct. Let me re-read carefully. 'Limit(4*n/(2*n + 4), n, oo, dir='-') = 2'. This is the limit of the ratio a_n/b_n where b_n=1/n. The text says 'The limit is finite and positive, so the two series do the same thing.' This is the correct application of LCT. Why would it fail? Maybe the 'dir='-' is weird syntax but the result is right. Is there a subtle error? No, the math is sound. Wait, look at step 1: 'For large n the term behaves like 1/n^1'. This is heuristic. Step 2 does the formal LCT. Step 3 concludes. It seems correct. Let me check for 'misleading'. The notation 'Limit(4*n/(2*n + 4)...)' is the limit of the ratio, not the term. The sentence says 'compare with that p-series'. It doesn't explicitly state b_n=1/n, but it's implied. Is it an error to not define b_n? It's slightly loose but standard. Let's look closer. Is it possible the user meant Direct Comparison? No, LCT is used. Is the limit calculation correct? 4n/(2n+4) -> 2. Yes. Does 1/n diverge? Yes. So the series diverges. The solution is correct. Why did I think it might fail? Maybe the 'dir='-' is a distractor. Or maybe the prompt implies I should find a fault. Let's look at the sentence 'The limit is finite and positive, so the two series do the same thing.' This is the conclusion of LCT. It is correct. I will mark it as pass.
Every verdict on record (4)
  • qwen3.6:27b-mlx: fail (error) 2026-10-05 — The Limit Comparison Test requires comparing the original term a_n = 4/(2n+4) with b_n = 1/n. The solution incorrectly computes the limit of 4n/(2n+4), which corresponds to comparing 4n with 1/n (or a_n with 1/n^2 if interpreted differently), rather than the ratio a_n/b_n = (4/(2n+4))/(1/n) = 4n/(2n+4). While the computed limit value 2 is correct for the ratio a_n/b_n, the sentence in step 1 claims the term behaves like 1/n, but step 2's equation label '4*n/(2*n + 4)' is ambiguous and technically represents the numerator of the ratio if b_n=1/n, but usually LCT is stated as lim a_n/b_n. The main issue is that step 1 says 'behaves like 1/n^1' but the equation in step 2 is just the limit of 4n/(2n+4). If b_n = 1/n, then a_n/b_n = 4n/(2n+4). The limit is 2. This is correct. However, the phrasing 'Limit(4*n/(2*n + 4)...)' is not the standard way to write the LCT limit, which is lim (a_n / b_n). It looks like the student might have confused the term itself with the ratio. But wait, if a_n = 4/(2n+4) and b_n = 1/n, then a_n/b_n = 4n/(2n+4). So the equation is correct for the LCT ratio. The conclusion is correct. Is there an error? Step 1 says 'compare with that p-series'. Step 2 calculates the limit of the ratio. Step 3 concludes divergence. This seems correct. Let me re-read carefully. 'Limit(4*n/(2*n + 4), n, oo, dir='-') = 2'. This is the limit of the ratio a_n/b_n where b_n=1/n. The text says 'The limit is finite and positive, so the two series do the same thing.' This is the correct application of LCT. Why would it fail? Maybe the 'dir='-' is weird syntax but the result is right. Is there a subtle error? No, the math is sound. Wait, look at step 1: 'For large n the term behaves like 1/n^1'. This is heuristic. Step 2 does the formal LCT. Step 3 concludes. It seems correct. Let me check for 'misleading'. The notation 'Limit(4*n/(2*n + 4)...)' is the limit of the ratio, not the term. The sentence says 'compare with that p-series'. It doesn't explicitly state b_n=1/n, but it's implied. Is it an error to not define b_n? It's slightly loose but standard. Let's look closer. Is it possible the user meant Direct Comparison? No, LCT is used. Is the limit calculation correct? 4n/(2n+4) -> 2. Yes. Does 1/n diverge? Yes. So the series diverges. The solution is correct. Why did I think it might fail? Maybe the 'dir='-' is a distractor. Or maybe the prompt implies I should find a fault. Let's look at the sentence 'The limit is finite and positive, so the two series do the same thing.' This is the conclusion of LCT. It is correct. I will mark it as pass.
  • gpt-oss:20b: pass 2026-10-05
  • qwen3.6:27b-mlx: pass 2026-10-05
  • gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.

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