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Absolute and conditional convergence

Problem 7.247 · medium

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{2 n + 1} \) converge absolutely, converge conditionally, or diverge?

\[ \lim_{n \to \infty} \frac{1}{2 n + 1} = 0 \]
bₙ → 0.✓ Proved
bₙ is eventually decreasing, so the alternating series test gives convergence.
\[ \lim_{n \to \infty}\left(\frac{n}{2 n + 1}\right) = \frac{1}{2} \]
Limit comparison of |aₙ| with 1/n^1: a p-series that diverges.✓ Proved

Answer \( \text{converges conditionally} \)

Lines: 2 proved, 1 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	SymPy's Sum.is_convergent on the series and on its absolute values

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution claims the limit comparison test with 1/n shows divergence of the absolute series, but the limit of |a_n|/(1/n) is 1/2, not 1/2 * 1 (wait, limit is 1/2, which is positive and finite, so the test is valid). However, the sentence in step 2 is incomplete: it states 'b_n is eventually decreasing' but does not verify this condition or cite that b_n > 0. More critically, step 3 calculates the limit of n/(2n+1) which is 1/2, implying |a_n| ~ 1/(2n). The limit comparison test requires comparing |a_n| to 1/n. The limit is lim |a_n| / (1/n) = lim n/(2n+1) = 1/2. Since 1/2 > 0, and sum 1/n diverges, sum |a_n| diverges. This part is actually correct. Let's re-read carefully. Step 2 says 'b_n is eventually decreasing'. It does not explicitly state b_n > 0, though it is implied. Is that an error? Usually, AST requires b_n > 0 and decreasing. The solution skips the positivity check. Is that a 'style' or 'error'? In rigorous calculus, skipping a hypothesis is an error. Also, the phrasing 'Limit comparison of |a_n| with 1/n^1' is slightly awkward but mathematically sound. The main issue is Step 2: it asserts 'b_n is eventually decreasing' without proof or justification, and omits the positivity condition. While 'eventually decreasing' is true, a solution should ideally verify it or at least mention positivity. However, looking at the prompt's constraints: 'Check that... each theorem or test is applied where its hypotheses hold'. The AST requires b_n > 0 and b_n decreasing. The solution only mentions decreasing. It fails to state b_n > 0. This is a missing hypothesis. Therefore, it is an error.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims the limit comparison test with 1/n shows divergence of the absolute series, but the limit of |a_n|/(1/n) is 1/2, not 1/2 * 1 (wait, limit is 1/2, which is positive and finite, so the test is valid). However, the sentence in step 2 is incomplete: it states 'b_n is eventually decreasing' but does not verify this condition or cite that b_n > 0. More critically, step 3 calculates the limit of n/(2n+1) which is 1/2, implying |a_n| ~ 1/(2n). The limit comparison test requires comparing |a_n| to 1/n. The limit is lim |a_n| / (1/n) = lim n/(2n+1) = 1/2. Since 1/2 > 0, and sum 1/n diverges, sum |a_n| diverges. This part is actually correct. Let's re-read carefully. Step 2 says 'b_n is eventually decreasing'. It does not explicitly state b_n > 0, though it is implied. Is that an error? Usually, AST requires b_n > 0 and decreasing. The solution skips the positivity check. Is that a 'style' or 'error'? In rigorous calculus, skipping a hypothesis is an error. Also, the phrasing 'Limit comparison of |a_n| with 1/n^1' is slightly awkward but mathematically sound. The main issue is Step 2: it asserts 'b_n is eventually decreasing' without proof or justification, and omits the positivity condition. While 'eventually decreasing' is true, a solution should ideally verify it or at least mention positivity. However, looking at the prompt's constraints: 'Check that... each theorem or test is applied where its hypotheses hold'. The AST requires b_n > 0 and b_n decreasing. The solution only mentions decreasing. It fails to state b_n > 0. This is a missing hypothesis. Therefore, it is an error.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Alternating Series Test for convergence and the Limit Comparison Test with the harmonic series to show divergence of the absolute series, leading to the correct conclusion of conditional convergence.
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/alternating_series, checked 2026-10-04 with SymPy 1.14.0.