Geometric series
Problem 7.172 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} - 2 \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ -2 \]The first term, a.✓ Proved
- \[ -1 \]The common ratio, r.✓ Proved
- |r| = 1 ≥ 1, so the series diverges.
Answer \( \text{diverges} \)
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 | ✓ 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 | the terms do not shrink (60th term checked) |
Reviewers
gpt-oss:20b: fail (misleading) — The solution incorrectly applies the ratio test: |r|=1 is inconclusive, yet it concludes divergence. A correct argument would note that the terms do not approach 0, so the series diverges.qwen3.6:27b-mlx: pass — The solution correctly identifies the series as geometric with |r|=1 and applies the divergence test for geometric series. The conclusion is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the series as geometric with |r|=1 and applies the divergence test for geometric series. The conclusion is correct.gpt-oss:20b: fail (misleading) 2026-09-28 — The solution incorrectly applies the ratio test: |r|=1 is inconclusive, yet it concludes divergence. A correct argument would note that the terms do not approach 0, so the series diverges.qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the common ratio as -1. The terms are 2, -2, 2, -2..., so the ratio is -1, but the first term is 2, not -2. Moregpt-oss:20b: pass 2026-09-28
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/geometric_series, checked 2026-09-28 with SymPy 1.14.0.