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Geometric series

Problem 7.154 · easy

Does \( \displaystyle \sum_{n=0}^{\infty} - 3 \left(-1\right)^{n} \) converge? If so, find its sum.

\[ -3 \]
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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the common ratio as -1. The terms are 3, -3, 3, -3..., so the ratio is -1, but the first term is 3 (or -3 depend

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the common ratio as -1. The terms are 3, -3, 3, -3..., so the ratio is -1, but the first term is 3 (or -3 depend
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the common ratio as -1. The terms are 3, -3, 3, -3..., so the ratio is -1, but the first term is 3, not -3. More
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.