Geometric series
Problem 7.202 · easy Video
Does \( \displaystyle \sum_{n=1}^{\infty} -2 \) converge? If so, find its sum.
- \[ -2 \]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 series is a constant series (-2 + -2 + ...), which is not a geometric series with r=1 (which would be 1 + 1 + ...). The correct reason for divergence is that the terms do not approach zero (n-th term test).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly identifies the series as geometric with common ratio r=1. The series is a constant series (-2 + -2 + ...), which is not a geometric series with r=1 (which would be 1 + 1 + ...). The correct reason for divergence is that the terms do not approach zero (n-th term test).gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution incorrectly identifies the series as geometric with ratio r=1. The series is actually a constant series (sum of -2), which diverges by the Divergence Test because the terms do not approach 0. While the final conclusion is correct, the reasoning is fundamentally flawed.gpt-oss:20b: fail (misleading) 2026-09-29 — The solution incorrectly treats the series as a geometric series with ratio r=1. The test |r|≥1 applies only to geometric series, but the given series is not geometric; its terms are constant -2, so the partial sums tend to -∞ and the series diverges. The conclusion is correct, but the reasoning is misleading.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The series is a constant term series, not a geometric series with ratio 1. Identifying the common ratio as 1 is incorrect; the terms do not form a geometric progression in the standard sense required for that test, and the divergence is due to the terms not approaching zero.gpt-oss:20b: pass 2026-09-29
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.