Geometric series
Problem 7.221 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} - 5 \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ -5 \]The first term, a.✓ Proved
- \[ -1 \]The common ratio, r.✓ Proved
- |r| = 1 ≥ 1, so the series diverges.Reviewed
Answer \( \text{diverges} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 | 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 series as geometric with ratio r = -1. Since |r| = 1, the geometric series test correctly concludes divergence. The logic is sound and the conclusion matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the series as geometric with ratio r = -1. Since |r| = 1, the geometric series test correctly concludes divergence. The logic is sound and the conclusion matches the stated answer.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the series as geometric with ratio r = -1. Since |r| = 1, the geometric series test correctly concludes divergence. The logic is sound and complete.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.