Geometric series
Problem 7.72 · easy Video
Does \( \displaystyle \sum_{n=1}^{\infty} - \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ -1 \]The first term, a.
- \[ -1 \]The common ratio, r.
- |r| = 1 ≥ 1, so the series diverges.
Answer \( \text{diverges} \)
Lines: 3 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 common ratio r = -1. The terms are actually constant (-1), so the ratio between consecutive terms is 1, not -1. While the conclusion (divergence) is correct, the reasoning is mathematically flawed because the series is not geometric.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly identifies the series as geometric with common ratio r = -1. The terms are actually constant (-1), so the ratio between consecutive terms is 1, not -1. While the conclusion (divergence) is correct, the reasoning is mathematically flawed because the series is not geometric.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] The series is not a geometric series; the terms are constant (-1), so the partial sums oscillate. Applying the geometric series ratio test is incorrect because the ratio between consecutive terms is not constant (it alternates between 1 and -1 if one considers the sign changes, or is undefined/constant -1 depending on interpretation, but the standard geometric series formula requires a constant r, which doesn't apply here in the way implied for convergence criteria). More simply, the term test (limit of terms is not 0) is the correct and sufficient reason for divergence.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly identifies the series as geometric with common ratio r = -1. The terms are all -1, so it is a constant series, not a geometric one with ratio -1. While the conclusion (divergence) is correct, the reasoning is flawed because the series does not have a common ratio of -1 (the ratio of consecutive terms is 1).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.