Absolute and conditional convergence
Problem 7.242 · medium
Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n + 1} \cdot 2^{- n} \) converge absolutely, converge conditionally, or diverge?
- \[ \lim_{n \to \infty} 2^{- n} = 0 \]bₙ → 0.✓ Proved
- bₙ is eventually decreasing, so the alternating series test gives convergence.Reviewed
- \[ \lim_{n \to \infty}\left(2^{n} 2^{- n - 1}\right) = \frac{1}{2} \]The ratio test on |aₙ|: the limit is below 1, so Σ|aₙ| converges.✓ Proved
Answer \( \text{converges absolutely} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies absolute convergence via the ratio test on the absolute values. Although the alternating series test is mentioned, the final conclusion is supported by the stronger condition of absolute convergence.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies absolute convergence via the ratio test on the absolute values. Although the alternating series test is mentioned, the final conclusion is supported by the stronger condition of absolute convergence.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies absolute convergence using the ratio test on the absolute values, which implies convergence. The mention of the alternating series test is redundant but not incorrect.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.