Absolute and conditional convergence
Problem 7.245 · medium
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n} \) converge absolutely, converge conditionally, or diverge?
- \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]bₙ → 0.✓ Proved
- bₙ is eventually decreasing, so the alternating series test gives convergence.
- \[ \lim_{n \to \infty} 1 = 1 \]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: passqwen3.6:27b-mlx: fail (error) — The solution claims the limit comparison test yields a limit of 1 for |a_n| compared to 1/n, but the limit of |a_n|/(1/n) is actually 1, not the limit of 1 shown in line 3. More importantly, line 3 is labeled as a 'Limit' calculation resulting in 1, but the text says 'Limit comparison... a p-series that diverges'. The limit comparison test requires comparing |a_n| to b_n=1/n. The limit is lim |(-1)^(n+1)/n| / (1/n) = 1. Since 1 > 0 and sum(1/n) diverges, sum(|a_n|) diverges. The equation in line 3 `Limit(1, n, oo, dir='-') = 1` is mathematically true but irrelevantly stated; it computes the limit of the constant 1, not the ratio of the series terms. While the conclusion is correct, the justification in line 3 is garbled and does not correctly represent the application of the Limit Comparison Test (which involves a ratio, not just the limit of 1). A student might be confused about what is being limited. However, the bigger issue is that line 3 is presented as a computer-checked equation `Limit(1, n, oo) = 1`, which is trivial and doesn't show the comparison. The sentence associated with it tries to do the work, but the equation provided is not the comparison limit. This is a 'misleading' or 'error' in presentation. Let's look closer. The prompt says 'Limit(1, n, oo, dir='-') = 1'. This is just the limit of the constant function 1. It does not show the limit of the ratio |a_n|/b_n. The sentence says 'Limit comparison of |a_n| with 1/n^1'. The equation should ideally be `Limit(|a_n|/(1/n), n, oo) = 1`. The provided equation is technically correct (1=1) but fails to model the step described. It's a 'style' or 'misleading' issue because it doesn't match the text. But wait, is it an error? The text says 'Limit comparison...'. The equation shows `Limit(1...)`. It's a mismatch. I will mark it as misleading because it teaches that the limit comparison test is just checking if 1=1, rather than checking the limit of the ratio.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims the limit comparison test yields a limit of 1 for |a_n| compared to 1/n, but the limit of |a_n|/(1/n) is actually 1, not the limit of 1 shown in line 3. More importantly, line 3 is labeled as a 'Limit' calculation resulting in 1, but the text says 'Limit comparison... a p-series that diverges'. The limit comparison test requires comparing |a_n| to b_n=1/n. The limit is lim |(-1)^(n+1)/n| / (1/n) = 1. Since 1 > 0 and sum(1/n) diverges, sum(|a_n|) diverges. The equation in line 3 `Limit(1, n, oo, dir='-') = 1` is mathematically true but irrelevantly stated; it computes the limit of the constant 1, not the ratio of the series terms. While the conclusion is correct, the justification in line 3 is garbled and does not correctly represent the application of the Limit Comparison Test (which involves a ratio, not just the limit of 1). A student might be confused about what is being limited. However, the bigger issue is that line 3 is presented as a computer-checked equation `Limit(1, n, oo) = 1`, which is trivial and doesn't show the comparison. The sentence associated with it tries to do the work, but the equation provided is not the comparison limit. This is a 'misleading' or 'error' in presentation. Let's look closer. The prompt says 'Limit(1, n, oo, dir='-') = 1'. This is just the limit of the constant function 1. It does not show the limit of the ratio |a_n|/b_n. The sentence says 'Limit comparison of |a_n| with 1/n^1'. The equation should ideally be `Limit(|a_n|/(1/n), n, oo) = 1`. The provided equation is technically correct (1=1) but fails to model the step described. It's a 'style' or 'misleading' issue because it doesn't match the text. But wait, is it an error? The text says 'Limit comparison...'. The equation shows `Limit(1...)`. It's a mismatch. I will mark it as misleading because it teaches that the limit comparison test is just checking if 1=1, rather than checking the limit of the ratio.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The solution claims 'Limit comparison of |aₙ| with 1/n^1' but provides the limit of 1, which is the result of a Direct Comparison or Ratio, not a Limit Comparison Test (which requires lim |aₙ|/bₙ). Furthermore, it fails to explicitly state that the series does not converge absolutely, which is required to conclude 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.