Geometric series
Problem 7.147 · medium
Does \( \displaystyle \sum_{n=1}^{\infty} - 5 \left(- \frac{2}{3}\right)^{n - 1} \) converge? If so, find its sum.
- \[ -5 \]The first term, a.✓ Proved
- \[ - \frac{2}{3} \]The common ratio, r.✓ Proved
- |r| = 2/3 < 1, so the series converges to a/(1 − r).Reviewed
- \[ -3 \]a/(1 − r).✓ Proved
Answer \( -3 \)
✓ Nihil obstat Lines: 3 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the 400th partial sum agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the first term and common ratio, applies the geometric series convergence test and sum formula correctly, and arrives at the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the first term and common ratio, applies the geometric series convergence test and sum formula correctly, and arrives at the correct result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the first term and common ratio, applies the geometric series convergence test and sum formula correctly, and arrives at the correct result.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.