Geometric series
Problem 7.173 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ 1 \]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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the series as geometric with ratio r = -1. While the terms alternate between 1 and -1, the ratio test is inconclusive for |r|=1, and the series actually diverges by the Divergence Test (terms do not approach 0), not because it is a divergent geometric series in the standard summation sense.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly identifies the series as geometric with ratio r = -1. While the terms alternate between 1 and -1, the ratio test is inconclusive for |r|=1, and the series actually diverges by the Divergence Test (terms do not approach 0), not because it is a divergent geometric series in the standard summation sense.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the series as geometric with ratio r=-1. The terms are 1, -1, 1, -1..., so the ratio is indeed -1, but the divergpt-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.