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Geometric series

Problem 7.153 · medium

Does \( \displaystyle \sum_{n=1}^{\infty} - 2 \left(\frac{2}{3}\right)^{n - 1} \) converge? If so, find its sum.

\[ -2 \]
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
\[ -6 \]
a/(1 − r).✓ Proved

Answer \( -6 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the series as a geometric series, verifies the convergence condition |r| < 1, and applies the correct sum formula.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the series as a geometric series, verifies the convergence condition |r| < 1, and applies the correct sum formula.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the first term and common ratio, verifies the convergence condition for a geometric series, and applies the sum formula correctly.
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.