Geometric series
Problem 7.14 · easy
Does \( \displaystyle \sum_{n=0}^{\infty} - 4 \left(-2\right)^{n} \) converge? If so, find its sum.
- \[ -4 \]The first term, a.Reviewed
- \[ -2 \]The common ratio, r.Reviewed
- |r| = 2 ≥ 1, so the series diverges.Reviewed
Answer \( \text{diverges} \)
Lines: 3 reviewed. 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the common ratio and applies the geometric series divergence test. The conclusion is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the common ratio and applies the geometric series divergence test. The conclusion is sound.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The geometric series test states that a series diverges if |r| > 1. The condition |r| >= 1 is insufficient because the case |r| = 1 requires separate analysis (though it also diverges, the logic is imprecise). More importantly, the standard geometric series formula applies for |r| < 1; for |r| >= 1, the terms do not approach zero, so it diverges by the Divergence Test. While the conclusion is correct, the justification using |r| >= 1 as the direct criterion for divergence is technically loose and potentially misleading regarding the specific theorem conditions.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the common ratio and applies the geometric series divergence test. The conclusion is sound.gpt-oss:20b: pass 2026-09-26
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 2026-10-13 with SymPy 1.14.0.